# Strongly Principal Ideals of Rings with Involution

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 International Journal of Algebra, Vol. 2, 2008, no. 14, Strongly Principal Ideals of Rings with Involution Usama A. Aburawash and Wafaa M. Fakieh Department of Mathematics, Faculty of Science Alexandria University, Alexandria, Egypt Abstract The notion of strongly principal ideal groups for associative rings was introduced in [3] and its several properties were studied in [2] by using cyclic groups. Motivated by these concepts, we introduce here *-cyclic groups and strongly principal *-ideal ring groups for rings with involution and investigate their structural properties by attaching involution on their corresponding ground groups. Mathematics Subject Classification: 16W10 Keywords: groups *-cyclic groups,*-biideals, (strongly) principal *-ideal ring 1. Introduction Let P be a ring property. A group G is said to be P group, in the sense of Feigelstock and Schlussel [3] (see also [2]), if there exists a P ring R such that G = R +. A group G is said to be nil, again in the sense of the above cited references, in case the only ring R with G = R + is the zero ring. If G is not nil and every ring R, with R 2 0 and G = R is a P ring, then G is said to be strongly P group. By using these concepts, strongly principal ideals are thoroughly investigated. For rings with involution, we introduce here the notion of strongly principal *-ideal ring groups and study their structural properties. To achieve this goal and to make the calculations simpler we have introduced *-cyclic groups, which is generated by an element a and its involutary image a in the group. Moreover, we study some structural properties of *-cyclic groups as well. In particular a formula (Corollary 3.2) for computing

2 686 U. A. Aburawash and W. M. Fakieh the number of involutions for abelian groups is obtained. Their direct sum and direct summand propert ies are also outlined. We have used *-cyclic groups to obtain various properties and classifications of (strongly) principal *-ideal ring groups. In particular, it is noticed in Theorem 4.7 that there exists no mixed strongly principal *-ideal ring groups. Throughout we assume that all groups are additive abelian and all rings are associative. If R is a ring, then its underlying additive group is denoted by G = R +. If x R, then x (respectively (x)) means the ideal of R (respectively the subgroup of G) generated by x. A ring R (respectively a group G) together with a unary operation is said to be a ring (respectively, a group) with involution in case for all a, b R, (a ) = a, (a + b) = a + b, and (ab) = b a (respectively, for all a, b G, (a ) = a, and (a + b) = a + b ).Thus the involution on R is an antiisomorphism of order two. For commutative rings, the identity mapping is clearly an involution. Nevertheless, every group has at least one involution, namely, the unary operation of taking inverse; that is g = g for every g G. Let a group G be decomposed into its subgroups as G = H K. If G has an involution, then is said to be changeless involution in case g =(h,k ), g =(h, k) H K (see[1]). A group G is said to be *-cyclic if for some a G, G =(a)+(a ), which indeed one may rewrite as G =(a) =(a, a ). Clearly, every cyclic group is *-cyclic, but the converse is not true in general (see an example in Section 3). A nonzero ideal I of an involution ring R (a nonzero subgroup H of an involution group G) which is closed under involution is termed as a *-ideal (I R)(respectively a *-subgroup (H G)); that is I = {a R a I } I. A subring A of R is said to be a biideal of R if ARA A and a *-biideal if, in addition, it is closed under the involution of R. A is called a principal *-biideal (see [7]), if A = a bi = Za + Za + ara + a Ra + ara + a Ra. On the same ground a principal *-ideal is defined. A principal *-ideal I is a *-ideal generated by a single element. This means that, for some a R, one may write: I = a = Za + Za + ar + Ra + RaR + a R + Ra + Ra R.

3 Strongly principal ideals of ringswith involution 687 Thus, it can easily be deduced that I = a + a = a, a. A ring with involution is said to be principal *-ideal ring if each *-ideal is a principal *-ideal. A groupg is called strongly principal *-ideal ring group, ifg is not nil and every ring R with involution satisfying R 2 0 and G = R +, is a principal *-ideal ring. Let f : A B be a group or a ring homomorphism. If A and B are equipped with some involutions A and B such that f(a A )=[f(a)] B, then we say that f is an involution preserved homomorphism. If f is an involution preserved isomorphism, then we will write A = B. It is clear that *-subgroups and *-ideals are preserved under such isomorphisms. Moreover, if A = B, as a group or a ring, then every involution on A induced an involution on B. In sections 2 and 3,we give some elementary properties for *-cyclic groups. Furthermore,in section 4,(strongly) principal *-ideal ring groups are widely studied. 2. Some Elementary Properties Lemma 2.1. Let G be a group with involution. Then the following subgroups of G are closed under the involution. (a) ng, n Z. (b) The torsion subgroup G t of G. (c) For any prime p, every p-primary subgroup G p of G. (d) The maximal divisible subgroup of G. (e) The subgroup G[m] ={g G mg =0} of G, for some integer m. Proof : (a) Let x ng. Then x = ng for some g G, hence x = ng and g G. So x ng and ng is closed under involution. (b) Let x G t. Then there exists a positive integer n such that nx =0. Hence nx = 0 and x G follows.. (c) Let x G p. Then x = p n for some positive integer n and p n x =0, implies p n x =0. Hence x G p. (d) Let D be the maximal divisible subgroup of G. If x D, then there exists y D such that x = ny for any positive integer n,whence x = ny. Since D is the maximal divisible subgroup, x,y D, therefore D is closed under involution. (e) Let x G[m], then mx = 0,whence mx = 0 and x G[m] follows. Corollary 2.2. In every involution ring R, nr, R[n], R t, R p and the maximal divisible ideal D are *-ideals.

4 688 U. A. Aburawash and W. M. Fakieh Proof : It is clear that G = R + has involution; the same involution of R. So from Lemma 2.1, nr, R[n], R t,r p and the maximal divisible subgroup D are *-subgroups of G. Since nr, R[n], R t,r p,and D are ideals in R (see [5]), hence nr, R[n], R t, R p and D are *-ideals in every involution ring R. Lemma 2.3. (a) Every direct sum of involution groups is an involution group. (b) Every direct summand of a group with a changeless involution is an involution group. (c) If a direct summand of a group has an involution, then the group has an involution. Proof : (a) Let G = H K, where H and K are groups with involutions H and K, respectively. Then for every g =(h, k) G, where h H and k K,define the involution G on G by g G =(h H,k K ). Because of the unique representation of each element, G becomes a unary operation on G. Further, (g G ) G =((h H ) H, (k K ) K )) = (h, k) =g. Assume that g i G, with g i =(h i,k i ), where h i H and k i K. Then (g 1 + g 2 ) G = ((h 1 + h 2 ) H, (k 1 + k 2 ) K )=((h H 1 + h H 2 ), (k K 1 + k K 2 )) = (h H 1,k K 1 )+(h H 2,k K 2 )=g G 1 + g G 2. Hence G is an involution on G ;it is in fact the changeless involution on G. The proof can analogously be extended to finite as well as to arbitrary direct sums. (b) Let G = H K. Set H = H 0 and K =0 K. Clearly, H and K are direct summands and subgroups of G. Assume that is the changeless involution on G. Then H (involution on G restricted to H ) is an involution on H and K is an involution on K. Also, H = H and K = K. Hence (b) is proved. (c) Let G = H K and H be a group with an involution H. Then for every g =(h, k) G, where h H and k K, define an operation G on G by g G =(h H,k). Clearly, G is the changeless involution on G.

5 Strongly principal ideals of ringswith involution 689 Parts (a ) and ( b) of Lemma 2.3 can easily be extended to rings, subrings and ideals. But for part (c) we need the following modification. Corollary 2.4. Let R = A B, where A and B are rings in which B is commutative. Then R has a (changeless) involution if and only if G has an involution. Proof : One way is clear from Lemma 2.3-(b). Assume that A has an involution A. Define R on R by r R =(a A,b) Then, R is a unary operation on R and for r 1 =(a 1,b 1 ),r 2 =(a 2,b 2 ) R, (r 1 r 2 ) R =((a 1 a 2 ) A,b 1 b 2 )=((a A 2 a A 1 ),b 2 b 1 )=r R 2 r R 1. The rest is as in Lemma 2.3-(c). 3. Cyclic Groups with Involution Let G be an infinite cyclic group. Following [8], there are two involutions on G, the identity involution and the involution a = a ; of taking inverse. If G is a finite cyclic group of order n, then Aut(G) consists of all automorphisms, α k : a ka, where 1 k n and (k, n) = 1. Moreover, Aut(G) = U (/Z n ) (the multiplicative group of units of the ring /Z n ). Since α n 1 : a (n 1) a is the only automorphism of order 2, Aut(G) has only two automorphisms of order two; the identity mapping and α n 1 (of taking inverse), so G has only two involutions. From this introduction,we note that every cyclic group has two involutions; namely the identity mapping and the mapping of taking inverse. Moreover, every subgroup of a cyclic group is closed under these involutions. Proposition 3.1. Let G be an additive abelian group, G = H K, and let H and K be cyclic subgroups of G. If( H, K ) 1, then (a) G has exactly four involutions, namely: g =(h, k), g =( h, k), g =(h, k), g =( h, k) and g =(h, k).

6 690 U. A. Aburawash and W. M. Fakieh (b) Every subgroup of G is closed under involution. Proof : (a) By Lemma 2.3, H and K are *-subgroups. Since H and K are cyclic, H and K,each, has two involutions; the identity involution and : a a. Hence again by Lemma 2.3, G has exactly the given four involutions. (b) By Theorem 8.1 in [4], any subgroup H of G is a direct sum of two cyclic subgroups, or it is cyclic. Hence by (a), H is a *-subgroup. The following immediate result gives the number of involutions of abelian groups. Corollary 3.2. Let G be an additive abelian group. If G = n H i, where each H i is a cyclic subgroup of G such that ( H i, H j ) 1, 1 i, j n, then G has 2 n involutions. Proposition 3.3. Let R be a ring with involution such that R + = G. Then R has only the identity involution in case any one of the following holds: (1) G is a cyclic group. (2) G is a direct sum of cyclic subgroups. Proof : (1) Let G be cyclic. Since R is an involution ring, G has either the identity involution or the involution : a a. However, (ab) ( b)( a), for all a, b R. Hence R has the identity involution only. (2) If G = H K, and H and K are cyclic subgroups of G, then by Proposition 3.1, G has four involutions. But then again by (1), G has only one involution. Definition 3.4. By a *-cyclic group H, we mean a *-group generated by one element. This means that, i=1 H =(a) =(a, a )=(a)+(a ). Let G be a cyclic group, then G =(a) =(a ) and G =(a)+(a ), so G is a *-cyclic group. The converse of this fact is not always true. For example in the group (M 2 2 (Z 3 ), +) ( ) 0 1 with the transposed involution, let a =, whence a 0 0 = Obviously, the *-cyclic group H =(a)+(a ) is not a cyclic group. ( ).

7 Strongly principal ideals of ringswith involution 691 Proposition 3.5. Let G = H K. IfH and K are *-cyclic groups such that H =(a), K =(b), ( a, b ) =1. Then G is *-cyclic. Proof : The given condition ( a, b ) =1 implies that (a) (b) is a cyclic group generated by (a, b) and (a ) (b )isa cyclic group generated by (a,b ). But, G = H K =(a)+(a ) (b)+(b )=(a) (b)+(a ) (b ). Hence G is *-cyclic with G =((a, b)). Proposition 3.6. If G is a *-cyclic group, then any *-subgroup of G is a *-cyclic subgroup. Proof : A *-cyclic group is either torsion or torsion free.first assume that G is torsion free and let If G =(a) =(a, a ). (a) (a ) 0, then na = ma 0, for some integers m and n. This implies na = ma. So, na na = ma ma, from which n(a a )=m(a a) = m(a a ) and so (n + m)(a a )=0. Since G is torsion free, a a = 0 implies a = a,whence (a) (a ) = 0 and G =(a) (a ). Secondly assume that G is torsion, G =(a)+(a ), and a = a = k.let g G, g = ma + na, for some integers m, n. Since k(ma + na ) = 0,it follows that g k, and a, a have maximal orders. Hence G =(a) (a ), from[6],page 81.Thus in both cases,g =(a) (a ). If H G, then H =(b) (c), where (b) (a) and (c) (a ). Then (b) =(ma) and (c) =(na ),whence H =(ma) (na ). Since H is *-subgroup, ma + na H. But ma na (ma), so n>m. Therefore n = m and (na ), so m>nand H =(na) (na ). Hence H =(na) and H is a *-cyclic subgroup of G.

8 692 U. A. Aburawash and W. M. Fakieh Proposition 3.7. Let x 1 and x 2 be elements of a group G such that a prime p x 1, x 2. If G =(x 1 ) (x 2 ), then there exist y 1, y 2 G such that (y 1 ) (x 1 ) and (y 2 ) (x 2 ). Proof : Let G =(x 1 )+(x 1 ) (x 2)+(x 2 ). If p is a prime such that p x 1, then there exists y 1 (x 1 ) such that, p y 1 and y 1 divides x 1. Consequently, Similarly there is y 2 (x 2 ) such that Hence, it is concluded that and (y 1 ) (x 1 ) and (y 1 ) (x 1 ). (y 2 ) (x 2 ) and (y 2 ) (x 2 ). (y 1 )+(y 1 ) (x 1)+(x 1 ) (y 2 )+(y 2 ) (x 2)+(x 2 ), that is, (y 1 ) (x 1 ) and (y 2 ) (x 2 ). 4. Strongly principal *-Ideal Ring Groups. As it is mentioned before that the notion of strongly principal ideal groups for associative rings was introduced and investigated thoroughly in [2] and [ 3]. Motivated by these concepts, we introduce here strongly principal *-ideal ring groups for rings with involution and study their structural properties by attaching involution on their corresponding ground groups. Definitions 4.1. Let R be a ring with involution. For some a R, one may write: I = a = Za + Za + ar + Ra + RaR + a R + Ra + Ra R. Clearly I is an ideal of R closed under involution and is called the principal *-ideal generated by a. One may deduce that I = a + a = a, a.

9 Strongly principal ideals of ringswith involution 693 A ring with ivolution is a principal *-ideal ring if each *-ideal is a principal *-ideal. Moreover, we say that a group G is strongly principal *-ideal ring group, ifg is not nil, and every ring R with involution satisfying R 2 0, and G = R +, is a principal *-ideal ring. Lemma 4.2. Let G = H K, H 0, K 0, be a strongly principal *-ideal ring group. Then H and K are either both *-cyclic or both nil. Proof : Suppose that H is not nil. Let S be a *-ring with S + = H and S 2 0 and let T be the zero ring on K. By Corollary 2.5, the ring direct sum R = S T is a ring with involution satisfying R + = G and R 2 0. Since T is a *-ideal in R, T = x. Clearly K = T + =(x). Therefore K is not nil. Interchanging the roles of H and K yields that H is *-cyclic. Corollary 4.3. Let G = H K, H 0, K 0, be a strongly principal *-ideal ring group. Then H and K are *-cyclic. Proof : It suffices to negate that H and K are both nil. Let R =(G, ) be a ring with involution satisfying R ) Suppose that R 2 K. There exist h 0 H, k 0 K, such that R = h 0 + k 0. Let h H, since h R, there exist integers n and m, and x R 2 such that However, x K, so h = n(h 0 + k 0 )+m(h 0 + k 0 ) + x. h = nh 0 + mh 0 and H is *-cyclic, contradicting the fact that H is nil. 2) Suppose that R 2 K. For all g 1,g 2 G, define g 1 g 2 = π H (g 1 g 2 ), where π H is the natural projection of G onto H. Since (g 1 g 2 ) =(π H (g 1 g 2 )) = π H (g 1 g 2 ) = π H (g 2 g 1)=g 2 g 1, hence S =(G, ) is a ring with involution satisfying S 2 H. The argument employed in (1) yields that K is *-cyclic which contradicts the fact that K is nil. Theorem 4.4. Let G 0 be a torsion group. If G is cyclic or G = (x 1 ) (x 2 ), where x 1 x 2, x 1 = x 2 = p is a prime, then G is strongly principal *-ideal ring group. Proof : Assume that either G is cyclic or as given in the hypothesis. Then by Proposition. 3.3, any ring R with R + = G has only the identity involution. By [2, Theorem 4.2.3],G is strongly principal *-ideal ring group.

10 694 U. A. Aburawash and W. M. Fakieh Theorem 4.5. Let G be a torsion strongly principal *-ideal ring group. Then G is *-cyclic group or G =(x 1 ) (x 2 ), with x i = p, a prime, where i =1, 2. Proof : Suppose that G is a strongly principal *-ideal ring group. Let G be indecomposable. Then by [2, Corollary 1.1.5], G = Z p n,pa prime, 1 n. Ifn =, then G is divisible by [2, Proposition.1.1.3] and so G is nil, by [2, Theorem 2.1.1], which is a contradiction. Hence G is cyclic and so *-cyclic. Next, suppose that G = H K, H 0,K 0, by Lemma 4.2, either H and K are both *-cyclic or both nil. If H and K are nil, then they are both divisible, so G is nil, by [2, Theorem 2.1.1] which is again a contradiction. Therefore G =(x 1 ) (x 2 ), with x i = n i, i =1, 2. If (n 1,n 2 ) = 1, then G is *-cyclic, by Proposition Otherwise, let p be a prime divisor of (n 1,n 2 ). Then by Proposition. 3.7, with G =(y 1 ) (y 2 ) H y i = p m i,i=1, 2, and 1 m 1 m 2. Since (y 1 ) (y 2 ) is neither * cyclic nor nil, H = 0 by Lemma 4.2. products The y i.y j = p m 2 1 y 2,y i.y j =0where i, j =1, 2, induce a *-ring structure R on G with R 2 0. Therefore, R = s 1 y 1 + s 2 y 2, s 1 and s 2 are integers. Every element x R has the form: x = k x s 1 y 1 +(k x s 2 + m x p m 2 1 )y 2 + k xs 1 y 1 +(k xs 2 + m xp m 2 1 )y 2 where k x, m x, k x, and m x are integers. In particular, y 1 = k y1 s 1 y 1, and y 2 =(k y2 s 2 + m y2 p m2 1 )y 2. Hence if m 2 > 1, then k y1 s 1 1(modp), and k y2 s 2 + m y2 p m2 1 1(mod p), which imply that p k y1 and p s 2. However k y1 s 2 + m y1 p m2 1 0(mod p). So either p k y1 or p s 2 which is a contradiction. Therefore m 1 = m 2 =1.

11 Strongly principal ideals of ringswith involution 695 Theorem 4.6. Let G be a torsion group which is either *- cyclic or G = (x 1 ) (x 2 ) with x i = p,a prime,where i =1, 2. Then for any *-ring R with R + = G, R is a principal *-ideal ring. Proof : By Proposition. 3.6, non trivial *-cyclic groups are clearly principal *-ideal ring groups. Let with G =(x 1 ) (x 2 ) x i = p, i =1, 2 and let R be a *-ring with R + = G and R 2 0. If I is a proper *-ideal in R, then I =1,p,orp 2, and so I is a *- ideal generated by one element, we may assume that R < x i >, i =1, 2. Hence <x i > + =(x j ) for i =1, 2. This implies the following three relations: x i x j = k i x i, 0 k i <p, if i = j, i =1, 2, either k 1 0ork 2 0, x i x j =0, if i j, i, j =1, 2, and x i x j =0, for all i, j =1, 2. Put I = x 1 + x 2. Suppose that k 1 0. Let r, s be integers such that rk 1 + sp = 1. Then rx 1 (x 1 + x 2 )=rk 1 x 1 =(1 sp)x 1 = x 1 and r(x 1 + x 2 ) x 1 = rk 1x 1 =(1 sp)x 1 = x 1. Hence x 1 I and x 1 I, so (x 1 + x 2 ) x 1 = x 2 I and (x 1 + x 2 ) x 1 = x 2 I.

12 696 U. A. Aburawash and W. M. Fakieh Therefore I = R. If k 2 0, then the above argument, reversing the roles of the indices 1, 2 again yields I = R. Theorem 4.7. There are no mixed strongly principal *-ideal ring groups. Proof : Let G be a mixed strongly principal *-ideal ring group. G is decomposable by [2, Corollary.1.1.5], so by Lemma 4.2, G = H K, H 0,K 0, with H and K both *-cyclic, or both nil. 1) Suppose that H and K are both nil. There are no mixed nil groups by [2, Theorem 2.1.1]. So, we may assume that H is a torsion group, and that K is torsion free. Let R be a *-ring with R + = G and R 2 0. Clearly H is a *-ideal in R and so H = h. Let h = m, then mh =0. By [2, Theorem 2.1.1], H is divisible, and therefore not bounded, a contradiction. 2) Suppose that H =(x) and K =(e) with x = n, and e =. The products x 2 = xe = ex = e x = ex = xe = x e = 0 and e 2 = ne induce a *-ring structure R on G satisfying R 2 0. Therefore there exist integers s and t such that R = sx + te. Every y R is of the form y = m y sx +(m y + u y n)te + m y sx +(m y + u y n)te, with m y, m y, u yand u y integers. In particular, (m e + u e n)t =1. Hence t = ±1. Therefore, m x + u x n = 0 and so n m x. However, x = m x sx =0, is a contradiction. Theorem 4.8. Let G be a torsion free strongly principal *-ideal ring group. Then G is either indecomposable, or is the direct sum of two nil groups. Proof : By Lemma 4.2. it suffices to negate that Suppose this is so, the products: G =(x 1 ) (x 2 ), x i 0,i=1, 2. x i x j =3x i and x i x j = 0 for i = j =1, 2, x i x j = x i x j =0, for i j induce a ring structure R on G with involution * satisfying R 2 0. Therefore there exist nonzero integers k 1,k 2 such that R = k 1 x 1 + k 2 x 2.

13 Strongly principal ideals of ringswith involution 697 Every x R is of the form: x =(r x +3s x )k 1 x 1 +(r x +3t x )k 2 x 2 +(r x +3s x )k 1x 1 +(r x +3t x )k 2x 2 where r x,r x,s x,s x,t x, t x are integers.from r x1 +3s x1 = ±1, it follows that r x1 ±1(mod 3). However, r x1 +3t x1 =0 implies r x1 0(mod 3), which is a contradiction. Lemma 4.9. Let G and H be torsion free groups with G = H. Then G is a strongly principal *-ideal ring group if and only if H is. Proof : Let f : H G be a *-isomorphism such that G is a strongly principal *-ideal ring group. Let R =(H, ) be a ring with involution such that R 2 0. The product g 1 g 2 = f(h 1 h 2 ), where, g 1 = f(h 1 ) and g 2 = f(h 2 ), for all g 1,g 2 G, induces a ring structure S =(G, ) with S 2 0. Then G is a group with involution. Since (g 1 g 2 ) =[f(h 1 h 2 )] = f(h 1 h 2 ) = f(h 2 h 1 )=g 2 g 1. Hence S has an involution. Let I R, then f(i) S and there exists g G such that f(i) = g, with g = f(h), h H. We claim that I = h. Clearly h I. Let x I, then f(x) g and so f(x) =ng + mg + g y 1 + g y 2 + z 1 g + z 2 g where m, n are integers and y 1 = f( h 1 ),y 2 = f (h 2 ),z 1 = f (h 1) and z 2 = f (h 2) G.Thus f (x) = nf (h )+mf (h)+f (h h 1 )+f (h h 2 ) +f (h 1 h) +f (h 2 h ) = f (nh + mh + h h 1 + h h 2 + h 1 h + h 2 h )

14 698 U. A. Aburawash and W. M. Fakieh which concludes that x h. Hence I = h. Theorem 4.10: Let G be a torsion group. Then the following are equivalent: (1) G is bounded (2) G is a principal *-ideal ring group. Proof : (1) (2): Suppose that ng = 0 and n is a positive integer. Then [ ] G = Z p k p n α k where p is a prime with p k n and α k a cardinal number,by [2, Proposition 1.1.9]. For each p k n, put Then H p k = α k Z p k. G = H p k, p k n and there exists a commutative principal ideal ring R p k with unity and R + P k = H p k for all p k n,by [5, Lemma 122.3]. The ring direct sum R = R p k p k n is a principal *-ideal ring with the identity involution satisfying R + = G and R 2 0. (2) (1): Let R be a principal *-ideal ring with R + = G. Then R = x and n = x. SonG =0. Theorem 4.11: Let G be a mixed group. Then (1) If G is a principal *-ideal ring group, then G t is bounded and G/G t is a principal *-ideal ring group. (2) Conversely, if G t is bounded and if there exists a unital principal *-ideal ring with additive group G/G t, then G is a principal *- ideal ring group. Proof : (1) Let R be a principal *-ideal ring with R + = G. Since G t is a *-ideal in R, G t = x and ng t =0,n = x. Now G = G t H and H = G/G t by [2, Proposition ]. Now R = a + y,a G t, 0 y H.

15 Strongly principal ideals of ringswith involution 699 Suppose that R 2 G t and let h H. Then there exist integers k n, k n such that, h = k n y + k n y + b, with b R 2. Since R 2 G t,b=0, and h = k n y + k n y. Therefore H =(y). By Proposition 3.6, H is a principal *- ideal ring group. If R 2 G t, then R = R/G t is a principal *-ideal ring with R + = G/Gt, and R 2 0. (2) Conversely, suppose that G t is bounded, and that there exists a unital principal *-ideal ring T with T + = G/G t. Hence G = G t G/G t, by [2, Proposition ]. There exists a principal *-ideal ring S with unity and * is the identity involution such that S + = G t, from [5, Lemma 122.3]. Let R = S T with e, f the unities of S and T, respectively. Then R is a ring with involution *, by Corollary 2.5. Let I be a *-ideal in R, then Now, I S S and so Similarly Clearly, However, and I =(I S) (I T ). I S = x. I T =< y>. x + y I. x = e(x + y) x + y,x = e(x + y) x + y y = f(x + y) x + y, y = f(x + y) x + y. Hence we conclude that I =< x+ y>.

16 700 U. A. Aburawash and W. M. Fakieh References [1] U.A. Aburawash, On involution rings,east-west J.Math,Vol. 2, No. 2, (2000), [2] S. Feigelstock, Additive Groups of Rings, Pitman (APP), [3] S. Feigelstock, Z. Schlussel, Principal ideal and noetherian groups, Pac. J. Math, 75 (1978), [4] L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York, [5] L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, New York, [6] T.W. Hungerford, Algebra, Springer Science+Business Media, [7] N.V. Loi, On the structure of semiprime involution rings, Contr. to General Algebra, Proc. Krems Cons., North-Holland (1990), [8] D.J.S. Robinson, A Course in the Theory of Groups, Springer, New York, Received: December 25, 2007

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Algebra. Sample Solutions for Test 1

EPFL - Section de Mathématiques Algebra Fall semester 2008-2009 Sample Solutions for Test 1 Question 1 (english, 30 points) 1) Let n 11 13 17. Find the number of units of the ring Z/nZ. 2) Consider the

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS. 1. Ideals in Commutative Rings In this section all groups and rings will be commutative.

ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS PETE L. CLARK 1. Ideals in Commutative Rings In this section all groups and rings will be commutative. 1.1. Basic definitions and examples. Let R be a (commutative!)

### 3.1 The Definition and Some Basic Properties. We identify the natural class of integral domains in which unique factorization of ideals is possible.

Chapter 3 Dedekind Domains 3.1 The Definition and Some Basic Properties We identify the natural class of integral domains in which unique factorization of ideals is possible. 3.1.1 Definition A Dedekind

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Integral Domains. As always in this course, a ring R is understood to be a commutative ring with unity.

Integral Domains As always in this course, a ring R is understood to be a commutative ring with unity. 1 First definitions and properties Definition 1.1. Let R be a ring. A divisor of zero or zero divisor

### 1. A nonzero ideal in a principal ideal domain is maximal if and only if it is prime.

Hungerford: Algebra III.3. Factorization in Commutative Rings 1. A nonzero ideal in a principal ideal domain is maximal if and only if it is prime. Proof: Let R be a PID and let I be a proper ideal in

### COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:

COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative

### 2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H.

Math 307 Abstract Algebra Sample final examination questions with solutions 1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40, Determine H. Solution. Since gcd(18,

### Rough Semi Prime Ideals and Rough Bi-Ideals in Rings

Int Jr. of Mathematical Sciences & Applications Vol. 4, No.1, January-June 2014 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com Rough Semi Prime Ideals and Rough Bi-Ideals in

### 26 Ideals and Quotient Rings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

### (a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9

Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned

### Solutions to Homework Problems from Chapter 3

Solutions to Homework Problems from Chapter 3 31 311 The following subsets of Z (with ordinary addition and multiplication satisfy all but one of the axioms for a ring In each case, which axiom fails (a

### (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 2; (1, 1) : order 4; (1, 2) : order 2; (1, 3) : order 4.

11.01 List the elements of Z 2 Z 4. Find the order of each of the elements is this group cyclic? Solution: The elements of Z 2 Z 4 are: (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### 4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

### Set Ideal Topological Spaces. W. B. Vasantha Kandasamy Florentin Smarandache

Set Ideal Topological Spaces W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free:

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Cancelling an a on the left and a b on the right, we get b + a = a + b, which is what we want. Note the following identity.

15. Basic Properties of Rings We first prove some standard results about rings. Lemma 15.1. Let R be a ring and let a and b be elements of R. Then (1) a0 = 0a = 0. (2) a( b) = ( a)b = (ab). Proof. Let

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### ARITHMETIC PROGRESSIONS OF THREE SQUARES

ARITHMETIC PROGRESSIONS OF THREE SQUARES KEITH CONRAD 1 Introduction Here are the first 10 perfect squares (ignoring 0): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 In this list is the arithmetic progression

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Introduction to finite fields

Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field

Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field 1. Throughout this section, F is a field and F [x] is the ring of polynomials with coefficients in F. We will

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS

ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS T. BRAND, M. JAMESON, M. MCGOWAN, AND J.D. MCKEEL Abstract. For a commutative ring R, we can form the zero-divisor graph Γ(R) or the ideal-divisor

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

### MATH 251 Homework 7 Solutions

MATH 51 Homework 7 Solutions 1. Let R be a ring and {S j j J} a collection of subrings resp. ideals of R. Prove that j J S j is a subring resp. ideal of R. Proof : Put S j J S j. Suppose that S j is a

### The Structure of Galois Algebras

The Structure of Galois Algebras George Szeto Department of Mathematics, Bradley University Peoria, Illinois 61625 { U.S.A. Email: szeto@hilltop.bradley.edu and Lianyong Xue Department of Mathematics,

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### MATH 436 Notes: Examples of Rings.

MATH 436 Notes: Examples of Rings. Jonathan Pakianathan November 20, 2003 1 Formal power series and polynomials Let R be a ring. We will now define the ring of formal power series on a variable x with

### Computer Algebra for Computer Engineers

p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions

### NOTES ON DEDEKIND RINGS

NOTES ON DEDEKIND RINGS J. P. MAY Contents 1. Fractional ideals 1 2. The definition of Dedekind rings and DVR s 2 3. Characterizations and properties of DVR s 3 4. Completions of discrete valuation rings

### MATH PROBLEMS, WITH SOLUTIONS

MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These

### GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS

GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS GUSTAVO A. FERNÁNDEZ-ALCOBER AND ALEXANDER MORETÓ Abstract. We study the finite groups G for which the set cd(g) of irreducible complex

### FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set

FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,

### RINGS OF ZERO-DIVISORS

RINGS OF ZERO-DIVISORS P. M. COHN 1. Introduction. A well known theorem of algebra states that any integral domain can be embedded in a field. More generally [2, p. 39 ff. ], any commutative ring R with

### 4.1 Modules, Homomorphisms, and Exact Sequences

Chapter 4 Modules We always assume that R is a ring with unity 1 R. 4.1 Modules, Homomorphisms, and Exact Sequences A fundamental example of groups is the symmetric group S Ω on a set Ω. By Cayley s Theorem,

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

3. QUADRATIC CONGRUENCES 3.1. Quadratics Over a Finite Field We re all familiar with the quadratic equation in the context of real or complex numbers. The formula for the solutions to ax + bx + c = 0 (where

### Factorization in Polynomial Rings

Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18 of Gallian s Contemporary Abstract Algebra. Most of the important

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### 6 Commutators and the derived series. [x,y] = xyx 1 y 1.

6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS

International Electronic Journal of Algebra Volume 6 (2009) 95-106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### ZORN S LEMMA AND SOME APPLICATIONS

ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### F1.3YE2/F1.3YK3 ALGEBRA AND ANALYSIS. Part 2: ALGEBRA. RINGS AND FIELDS

F1.3YE2/F1.3YK3 ALGEBRA AND ANALYSIS Part 2: ALGEBRA. RINGS AND FIELDS LECTURE NOTES AND EXERCISES Contents 1 Revision of Group Theory 3 1.1 Introduction................................. 3 1.2 Binary Operations.............................

### DISTRIBUTIVE PROPERTIES OF ADDITION OVER MULTIPLICATION OF IDEMPOTENT MATRICES

J. Appl. Math. & Informatics Vol. 29(2011), No. 5-6, pp. 1603-1608 Website: http://www.kcam.biz DISTRIBUTIVE PROPERTIES OF ADDITION OVER MULTIPLICATION OF IDEMPOTENT MATRICES WIWAT WANICHARPICHAT Abstract.

### 7. Some irreducible polynomials

7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

### Factorization Algorithms for Polynomials over Finite Fields

Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 2011-05-03 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is

### 12 Greatest Common Divisors. The Euclidean Algorithm

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

### DEFINABLE TYPES IN PRESBURGER ARITHMETIC

DEFINABLE TYPES IN PRESBURGER ARITHMETIC GABRIEL CONANT Abstract. We consider the first order theory of (Z, +,

### fg = f g. 3.1.1. Ideals. An ideal of R is a nonempty k-subspace I R closed under multiplication by elements of R:

30 3. RINGS, IDEALS, AND GRÖBNER BASES 3.1. Polynomial rings and ideals The main object of study in this section is a polynomial ring in a finite number of variables R = k[x 1,..., x n ], where k is an

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

### 1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### 3. Prime and maximal ideals. 3.1. Definitions and Examples.

COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

### Localization (Hungerford, Section 3.4)

Lemma. (1) If p is an ideal which is maximal with respect to the property that p is not finitely generated then p is prime. (2) If p is an ideal maximal with respect to the property that p is not principal,

### Algebraic Structures II

MAS 305 Algebraic Structures II Notes 12 Autumn 2006 Factorization in integral domains Lemma If a, b, c are elements of an integral domain R and ab = ac then either a = 0 R or b = c. Proof ab = ac a(b

### A Prime Ideal Principle in Commutative Algebra

A Prime Ideal Principle in Commutative Algebra T. Y. Lam and Manuel L. Reyes Abstract In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime.

### 4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express