ADDITIVE GROUPS OF RINGS WITH IDENTITY

Save this PDF as:

Size: px
Start display at page:

Transcription

1 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 rings with identity are studied. Results are also given for not reduced splitting mixed rings with identity. The Abelian groups G such that, excepting the zero-ring, every ring on G has identity are also determined. 1. Introduction In the sequel, all the groups we consider are nonzero Abelian, and all the rings are nonzero and associative. As customarily, for a ring R, we denote the additive (Abelian) group by R +, and for an Abelian group G, we say that R is a ring on G if R + = G. Hereafter, a group will be called an identity-group (identity for short) if there exists an associative ring with identity on G and strongly identity-group (Sidentity for short), if it is identity and, excepting the zero-ring, all associative rings on G have identity. A group G is called nil if the only ring on G is the zero-ring. Clearly nil groups are not identity nor S-identity. Since unital rings embed in the endomorphism rings of their additive groups it follows that identity-groups are isomorphic to additive groups of unital subrings of endomorphism groups. As a special case, endo-groups (and among these, additive groups of so-called E-rings) are identity-groups. Here an Abelian group G is called anendo-groupifthereisaringrovergsuchthatr = End(G), aringisomorphism. However, a simple comparison (see , [1]) shows that endo-group is far more restrictive than identity-group (e.g. any finitely generated group is identity, but only cyclic groups are endo-groups). It was known from long time that a torsion group is the additive group of a ring with identity if and only if it is bounded, i.e., a bounded direct sum of finite (co)cyclic groups. Consequently, it is not hard to show that the only S-identity torsion groups are the simple groups Z(p), for any prime number p (for a proof, see next Section). Moreover, the only torsion-free S-identity group is Q, the full rational group, and there are no mixed S-identity groups. Since there are no results on torsion-free or mixed identity-groups, this is what we investigate in this note. In the torsion-free case we obtain results when the typeset T(G) contains a minimum element which is idempotent and in the mixed case significant results are found in the case of splitting mixed groups which are not reduced. Date: date Mathematics Subject Classification. 20K99, 20K15, 20K21. Key words and phrases. additive group of ring, ring with identity, identity-group, Dorroh extension. 0

2 ADDITIVE GROUPS OF RINGS WITH IDENTITY 1 In the general case, the problem seems to be difficult (as many problems related to torsion-free or mixed rings are). Thus, this note also intends to open a possible new area of research. For definitions, notations and results on Abelian groups we refer to L. Fuchs [2]. The largest divisible subgroup of an Abelian group G will be called the divisible part of G, and a direct complement of the torsion part will be called a torsion-free part. For results on additive groups of rings we refer to S. Feigelstock [1]. The left annihilator of a ring R is ann l (R) = {a R ar = 0}. Similarly, the right and the annihilator of a ring are defined. Obviously, any ring with identity must have zero left (or right, or two-sided) annihilator. Since in any ring, disjoint ideals annihilate each other, in the additive group of a ring, different primary components annihilate each other, and any torsion-free ideal annihilates the torsion part. For a subgroup H of R +, (H) denotes the ideal generated by H in R. 2. Strongly identity groups Any ring multiplication may be extended from a direct summand, by taking the ring direct sum with the zero-ring on a direct complement (also called a trivial extension). Since a finite direct (product) sum of rings has identity if and only if each component has identity (the finite decomposition of 1 into central idempotents), the ring obtained by any trivial extension has no identity. Hence S-identity groups are indecomposable. The strongly identity groups are rare phenomenon. Theorem 1. A group is S-identity if and only if it is isomorphic to Z(p), for a prime number p, or isomorphic to Q. Proof. Since there are no mixed indecomposable groups, we expect to find only torsion or torsion-free S-identity groups. Let G be an identity group with multiplication denoted x y for x,y G. For a prime p, consider the multiplication (associative and both left and right distributive together with ) x y = p(x y). Since this ring must be the zero ring or an unital ring, pg = 0 or pg = G for every prime p. In the first case, since G is indecomposable (and so cocyclic) G = Z(p). In the second case, G is indecomposable divisible and so G = Z(p ) (impossible, because not identity) or G = Q. Conversely, these two groups are S-identity. Indeed, for every k = 1,...,p 1 the multiplication by k (on Z(p)) has identity: it is just the multiplicative inverse k 1 in the field Z(p). As for Q, recall that multiplications on Q are determined by nonzero squares a 2 of rational numbers ([2], p. 291). That is, for nonzero elements c,d Q there are nonzero rationals r,s such that c = ra and d = sa, and c d = (rs)a 2. Since a 0, 1 = a 1 a is the identity for this (commutative, associative and without zero divisors) multiplication. 3. Torsion-free identity rings It is readily checked that free groups and divisible torsion-free groups are identity (more: a torsion-free group is a field-group if and only if it is divisible; see [1]). Hence, if G = D(G) H, is a decomposition with the divisible part D(G) and a reduced direct complement H, using the ring direct product, it follows at once that G is an identity-group whenever H is so. For easy reference we mention here ([2], 123.2) a generalization of the Dorroh ring extension.

3 2 SIMION BREAZ AND GRIGORE CĂLUGĂREANU Proposition 2. Let R be an A-algebra, with A a commutative ring with identity. Then the ring R A with R + A = A+ R + and multiplication defined by (a 1,r 1 ) (a 2,r 2 ) = (a 1 a 2,a 1 r 2 +a 2 r 1 +r 1 r 2 ) for all a 1,a 2 A, r 1,r 2 R, is a ring with identity. The map R R A via r (0,r) is an embedding of R in R A as an ideal, and R A /R = A. Corollary 3. For any group G, the direct product (sum) Z G is an identity-group. Proof. It suffices to take the trivial multiplication on G. In the torsion free case we have a converse for this corollary: Proposition 4. Let X = {e,g i i I} be a maximal independent system in G. The following are equivalent: (1) the partial operation defined by e e = e, e g i = g i e = g i, g i g k = 0 extends to a (ring) multiplication with identity e on G; (2) the characteristic χ(e) is idempotent and minimum, and e is a direct summand of G. Proof. (1) (2) (i) In order to define e e = e, since χ(ab) χ(a)χ(b), we need χ(e) χ(e) 2, and so (by definition of the characteristics product) χ(e) = χ(e) 2, i.e., e has idempotent characteristics. Further, since we need, e a = a e = a for every a G, χ(a) χ(a)χ(e) = χ(e)χ(a) χ(e) shows that e has minimum characteristic. (ii) Denote by N = g i i I. Then N N = 0. Since e N = 0, it remains to prove that G = e +N. Let x G be arbitrary. Since every element depends on a maximal independent set, there are nonnegative integers n,k such that nx = ke + y, with y N. If d = (n,k), then d divides y and (1/d)y N (because N is pure). Thus, dividing nx = ke+y by d, we can suppose (n,k) = 1. Multiplying the dependence relation by y yields nx y = ky. If uk +vn = 1, then y = uky +vny = unx y +vny, and so n divides y. Since (n,k) = 1, and n divides y, it alsodivides the identity e. Thus, Gitself is n-divisible (indeed, for every g G, g = g e and n e imply n g) and x = (k/n)e+(1/n)y, so x e +N (again (1/n)y N, because N is pure). (2) (1) Note that by (i) and (ii) it follows that N = g i i I is a direct complement for e. Moreover, since the type of e is idempotent then e has a natural multiplication such that it becomes a ring isomorphic to a unital subring P Z of Q generated (as subring) by all 1 p with p P, where P is the set of all primes p with the property that e is p-divisible. Since N is a P Z-module, we can apply the previous proposition to obtain the conclusion. Remark. In the unital ring on G that occurs by this construction, 1 P 1 Z is the identity. Denoting it by e G, the multiplication (with identity e) on G is given by linearly extending (over P 1 Z) e e = e, e h = h e = h, h h 1 = 0 (h,h 1 H). Observe that the characteristic of χ(e) is idempotent and minimum in G. This way, torsion-free identity-groups which admit a Dorroh-like ring multiplication are characterized. Remark. In a torsion-free group, the subgroup purely generated by an element of idempotent and minimum characteristic might not be a direct summand. For

4 ADDITIVE GROUPS OF RINGS WITH IDENTITY 3 instance, consider two rank 1 nil groups H,K with types t(h) = (0,1,0,1,...) and t(k) = (1,0,1,0,...). Then the direct sum H K contains an element h + k of (minimum and idempotent) characteristics (0,0,...), with χ(h) = (0,1,0,...) and χ(k) = (1,0,1,...). But the subgroup purely generated by h + k is not a direct summand. For the case of mixed groups (by a mixed group we mean a genuine mixed group, i.e. 0 T(G) G), since there exist no nil mixed groups (Szele, [2]), when finding identity-groups, no mixed groups have to be excepted. Recall (4.6.3, [1]): let R be a ring with trivial left annihilator and R = A+B for subsets A,B of R. If A 2 = B 2 = 0 then R = 0. Hence, a (direct) sum of two (nonzero) nil groups is not identity and so, a mixed group with divisible torsion part and nil torsion-free complement is not identity. But nil torsion-free groups are far of being known, so this covers only a few known situations. We first settle the case when the torsion-free part is divisible. Proposition 5. A mixed group with divisible torsion-free part is identity if and only if the torsion part is bounded. Proof. Such groups are splitting and so have the form G = T(G) ( Q) with torsion part T(G). If R is a ring on G, it is proved in [1] (4.3.15), that any ring with trivial annihilator on the group direct sum above is also a ring direct sum on the components (i.e., it is fissible). Since identity rings have trivial annihilator, and ring direct sums have identity if and only if the components have identities, T(G) must be bounded. Conversely, D(G) = Q is known to be identity and so, if T(G) is bounded, G is identity. The second case we discuss is when the torsion part is not reduced. Proposition 6. If the torsion part is not reduced, i.e., G = DT(G) H with (maximal) torsion divisible DT(G), and R is a ring with identity on G, then (a) for any relevant prime relative to DT(G), H is not p-divisible and (b) h p (1) = 0 (in both H or G). Proof. (a) As intersection of two ideals (the torsion part and the divisible part are fully invariant subgroups), DT(G) is also an ideal which, being torsion divisible, must be nil (ideal), i.e., t t = t t = 0 for every t,t DT(G). If H is p-divisible then G = Z(p ) K and Z(p ) annihilates R and so G is not identity group. Indeed, let x R, a Z(p ) with ord(a) = p k. Since G is p-divisible, there is y G such that x = p k y and it is readily checked that a x = x a = 0. (b) For any relevant prime p (relative to DT(G)) we show that 1 / pg. Indeed, otherwise 1 pg implies pg = G, the left member being ideal in R, which is impossible: pg = p(t(g) H) = pt(g) ph = T(G) ph < G. The third case is with a divisible torsion part, again splitting, so let R be a ring on G = T(G) V, with reduced torsion-free V. Here we must except the (already discussed) identity case when V has a cyclic direct summand. If for a relevant prime p, V is p-divisible, then Z(p ) annihilates all R and so G is not identity.

5 4 SIMION BREAZ AND GRIGORE CĂLUGĂREANU The remaining case is when pv V for all relevant primes. Some groups of this type are considered in [1](p. 8). Suppose G = H Z(p ) with torsion-free not p-divisible H. Choose b H with h p (b) = 0. For any positive integer n, choose a n Z(p ) with ord(a n ) = p n. The map b b a n can be extended to an epimorphism H H Z(p n ). Therefore a ring R with R + = G can be constructed so that (H) Z(p ) is an arbitrary proper subgroup of Z(p ). However, these are not unital rings. Indeed, denote by R a ring constructed as above. Then R/(H) = (H)+Z(p ) = Z(p ) (H) Z(p n ) = Z(p ) is a divisible p-group, which (not being bounded) is not identity. Hence so is G. Finally, note that a consequence of Proposition 2 also gives mixed identitygroups, using the Dorroh-like construction. That is Corollary 7. Let P be a set of prime numbers and G a p-divisible group with zero p-components for all p P. Then the direct sum (P 1 Z) + G is an identitygroup with respect to the Dorroh-like multiplication defined by (q 1,x 1 ) (q 2,x 2 ) = (q 1.q 2,q 1 x 2 +q 2 x 1 ) for all q 1,q 2 P 1 Z, x 1,x 2 G. Here, for a set of prime numbers P, P P, P 1 Z denotes the unital subring of Q generated (as subring) by all 1 p with p P. Obviously (P 1 Z) + is p-divisible for all p P. In closing this paper here are two results also related to our subject Proposition 8. A group has only identity-subgroups if and only if it is a direct sum of a bounded group and a torsion-free identity-group which has only identitysubgroups. Proof. Suppose G has only identity-subgroups. Since its torsion part must be identity, T(G) (together with its subgroups) is bounded and so (Baer-Fomin) G is splitting: G = T(G) V. Here V, but also all its subgroups, must be identity and so, must have an element of minimum and idempotent type. The converse is obvious. Despite the fact that, having only identity-subgroups seems a strong condition on torsion-free groups, we were not able to give a useful characterization for such groups. Since the subgroups purely generated by elements must have idempotent type (otherwise these are nil and so not identity) such groups have only idempotent types in their typeset (more, these types have only finitely many nonzero components). Among finite rank torsion-free Butler groups (a Butler group is a homomorphic image of a completely decomposable group) the quotient divisible groups are exactly those which have only idempotent types (see [3]), 8.6). Proposition 9. A group has only identity quotient groups if and only if it is a direct sum of a bounded group and a finite rank free group. Proof. Suppose G has only identity quotient groups. If G has infinite rank, there is an epimorphism G Q/Z, a contradiction. Thus G has finite rank. If we take a torsion-free quotient of rank 1, this has idempotent type. If this is not free, again we can find a torsion divisible quotient, a

6 ADDITIVE GROUPS OF RINGS WITH IDENTITY 5 contradiction. Therefore, every rank 1 torsion-free quotient is free and so G (having finite rank) has the decomposition we claimed: G = T F with a finite rank free F. Finally, if some p-component is not bounded, again we can find an epimorphism to a torsion divisible group, a contradiction. Since bounded groups and free groups are identity-groups, the converse follows. References [1] Feigelstock S. Additive Groups of Rings. Pitman Research Notes in Mathematics 83, Pitman, Boston-London, [2] Fuchs L. Infinite Abelian Groups. vol. 2, New York, Academic Press, [3] Lady E. L. Finite Rank Torsion Free Modules Over Dedekind Domains. Unpublished Notes (1998). [4] Rédei L., Szele T. Die ringe ersten Ranges. Acta Sci. Math. Szeged 12A (1950), Babeş Bolyai University, Department of Mathematics, Kogalniceanu Str 1, , Cluj-Napoca, Romania address: and

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

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

Strongly Principal Ideals of Rings with Involution

International Journal of Algebra, Vol. 2, 2008, no. 14, 685-700 Strongly Principal Ideals of Rings with Involution Usama A. Aburawash and Wafaa M. Fakieh Department of Mathematics, Faculty of Science Alexandria

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

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

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

4.1 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,

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

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

(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

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

Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

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

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

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(

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

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

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

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

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

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

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

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

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

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

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

EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute

Commutative Algebra seminar talk

Commutative Algebra seminar talk Reeve Garrett May 22, 2015 1 Ultrafilters Definition 1.1 Given an infinite set W, a collection U of subsets of W which does not contain the empty set and is closed under

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

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

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,

Solutions A ring A is called a Boolean ring if x 2 = x for all x A.

1. A ring A is called a Boolean ring if x 2 = x for all x A. (a) Let E be a set and 2 E its power set. Show that a Boolean ring structure is defined on 2 E by setting AB = A B, and A + B = (A B c ) (B

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

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

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

r(x + y) =rx + ry; (r + s)x = rx + sx; r(sx) =(rs)x; 1x = x

Chapter 4 Module Fundamentals 4.1 Modules and Algebras 4.1.1 Definitions and Comments A vector space M over a field R is a set of objects called vectors, which can be added, subtracted and multiplied by

(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

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,

NOTES ON CATEGORIES AND FUNCTORS

NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category

The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

5.1 Commutative rings; Integral Domains

5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

MODULES OVER A PID KEITH CONRAD

MODULES OVER A PID KEITH CONRAD Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to K n for some n 0. When we replace the scalar field K with a commutative

D-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 5

D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 5 Dihedral groups, permutation groups, discrete subgroups of M 2, group actions 1. Write an explicit embedding of the dihedral group D n into the symmetric

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,

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

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

Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

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

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

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

(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

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

Algebra 2. Rings and fields. Finite fields. A.M. Cohen, H. Cuypers, H. Sterk. Algebra Interactive

2 Rings and fields A.M. Cohen, H. Cuypers, H. Sterk A.M. Cohen, H. Cuypers, H. Sterk 2 September 25, 2006 1 / 20 For p a prime number and f an irreducible polynomial of degree n in (Z/pZ)[X ], the quotient

Factoring of Prime Ideals in Extensions

Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree

Non-unique factorization of polynomials over residue class rings of the integers

Comm. Algebra 39(4) 2011, pp 1482 1490 Non-unique factorization of polynomials over residue class rings of the integers Christopher Frei and Sophie Frisch Abstract. We investigate non-unique factorization

1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

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

arxiv:math/ v1 [math.nt] 31 Mar 2002

arxiv:math/0204006v1 [math.nt] 31 Mar 2002 Additive number theory and the ring of quantum integers Melvyn B. Nathanson Department of Mathematics Lehman College (CUNY) Bronx, New York 10468 Email: nathansn@alpha.lehman.cuny.edu

INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number

Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and

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

EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.

ABSTRACT ALGEBRA. Romyar Sharifi

ABSTRACT ALGEBRA Romyar Sharifi Contents Introduction 7 Part 1. A First Course 11 Chapter 1. Set theory 13 1.1. Sets and functions 13 1.2. Relations 15 1.3. Binary operations 19 Chapter 2. Group theory

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

Define the set of rational numbers to be the set of equivalence classes under #.

Rational Numbers There are four standard arithmetic operations: addition, subtraction, multiplication, and division. Just as we took differences of natural numbers to represent integers, here the essence

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

Section IV.21. The Field of Quotients of an Integral Domain

IV.21 Field of Quotients 1 Section IV.21. The Field of Quotients of an Integral Domain Note. This section is a homage to the rational numbers! Just as we can start with the integers Z and then build the

FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS

FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure

Proof. The map. G n i. where d is the degree of D.

7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

Solutions to Assignment 4

Solutions to Assignment 4 Math 412, Winter 2003 3.1.18 Define a new addition and multiplication on Z y a a + 1 and a a + a, where the operations on the right-hand side off the equal signs are ordinary

G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

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

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

The Markov-Zariski topology of an infinite group

Mimar Sinan Güzel Sanatlar Üniversitesi Istanbul January 23, 2014 joint work with Daniele Toller and Dmitri Shakhmatov 1. Markov s problem 1 and 2 2. The three topologies on an infinite group 3. Problem

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

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

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

Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

PROBLEM SET 6: POLYNOMIALS

PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other

Galois representations with open image

Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group

Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

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