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, Bradley University Peoria, Illinois 61625 { U.S.A. Email: lxue@hilltop.bradley.edu Let B be a ring with 1 and G an automorphism group of B of order n for some integer n. It is shown that if B is a Galois algebra with Galois group G, then B is either a direct sum of central Galois algebras or a direct sum of central Galois algebras and a commutative Galois algebra. Moreover, when G is inner, B is either a direct sum of Azumaya projective group algebras or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. Examples are given for these structures. 1. INTRODUCTION T. Kanzaki ([6]), M. Harada ([4]), and F. R. DeMeyer ([3]) investigated Galois algebras. K. Sugano ([7]) studied the type of Galois H-separable extensions, and several other types of Galois Extensions were recently discussed ([1, 7, 8, 9, 10]). In [6], many interesting properties were found. It was shown that if B is a Galois algebra over a commutative 1

ring R with Galois group G, then B = Pg2G J g where J g = fb 2 B j bx = g(x)b for all x 2 Bg ([6, Theorem 1]), J g J h = I g J gh where I g = BJ g \ C and C the center of B, Ig 2 = I g, I g J g = J g ([6, Proposition 2]), and J g J g 1 = e g C for some idempotent e g in C ([6, Theorem 2]). In particular, if B is a central Galois algebra, then J g J h = J gh and I g = C ([6, Corollary 1]), and its converse was also shown by M. Harada ([4, Theorem 1]): if B = Pg2G J g such that J g J g 1 = C and B is a separable R-algebra, then B is a central Galois algebra. The purpose of the present paper is to show the following structure theorem for a Galois algebra over R: if B is a Galois algebra over R with Galois group G, then there exist orthogonal idempotents fe i 2 Cji = 1; 2; :::; m for some integer mg and some subgroups H i of G such that Be i is a central Galois algebra with Galois group H i for each i = 1; 2; :::; m and B = Pm i=1 Be i or B = ( Pm i=1 Be i) A for some commutative Galois algebra A with Galois group Gj A = G. When G is inner, B is either a direct sum of Azumaya projective group algebras as dened by F. R. DeMeyer ([3, Theorem 6]) or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. This paper was revised under the suggestions of the referee and written under the support of a Caterpillar Fellowship at Bradley University. We would like to thank the referee for the valuable suggestions and Caterpillar Inc. for the support. 2. DEFINITIONS AND NOTATIONS Throughout this paper, B will represent a ring with 1, G an automorphism group of B of order n for some integer n, C the center of B, and B G the set of elements in B xed under each element in G. We denote J g = fb 2 B j bx = g(x)b for all x 2 Bg and I g = BJ g \ C for each g 2 G. Let A be a subring of a ring B with the same identity 1. We call B a separable extension of A if there exist fa i ; b i in B, i = 1; 2; :::; m for some integer mg such that P ai b i = 1, and P ba i b i = P a i b i b for all b in B where is over A. An Azumaya 2

algebra is a separable extension of its center. B is called a Galois extension of B G with Galois group G if there exist elements fa i ; b i in B, i = 1; 2; :::; mg for some integer m such that P m i=1 a ig(b i ) = 1;g for each g 2 G. Such a set fa i ; b i g is called a G-Galois system for B. B is called a Galois algebra over R if B is a Galois extension of R which is contained in C, and B is called a central Galois extension if B is a Galois extension of C. A ring B is called a H-separable extension of A if B A B is isomorphic to a direct summand of a nite direct sum of B as a B-bimodule, and B is called a Galois H-separable extension if it is a Galois and a H-separable extension of B G (see [7]). 3. THE STRUCTURE THEOREM In this section, we shall show a structure theorem for a Galois algebra B over R with Galois group G. We begin with some properties of the C-module J g for g 2 G. LEMMA 3.1. Let B be a Galois algebra over R with Galois group G, then (1) B(J g J h ) = B(J h J g ) for all g; h 2 G. (2) BJg 2 = BJ g for all g 2 G. Proof. (1) Since B is a Galois algebra over R, R C and B is a separable extension of R. Hence B is an Azumaya C-algebra ([2, Theorem 3.8, page 55]). Noting that BJ g is an ideal of B and I g = BJ g \ C is an ideal of C, we have that BJ g = BI g ([2, Corollary 3.7, page 54]). Hence B(J g J h ) = (BJ g )(BJ h ) = (BI g )(BI h ) = (BI h )(BI g ) = (BJ h )(BJ g ) = B(J h J g ). (2) By Proposition 2 in [6], Ig 2 = I g for all g 2 G. Hence BJg 2 = (BJ g )(BJ g ) = (BI g )(BI g ) = BIg 2 = BI g = BJ g for all g 2 G. Lemma 3.2. Let B be a Galois algebra over R with Galois group G and H is a subgroup of G, then 3

g 2 H. (1) B( h2h J h ) is invariant under H, that is, g(b( h2h J h )) = B( h2h J h ) for each (2) B( h2h J h ) = Be for some idempotent e in C. (3) If h2h J h 6= f0g, Hj Be = H. Proof. (1) It is easy to check that g(j h ) = J ghg 1 for all g; h 2 G, so g(b h2h J h ) = B( h2h J ghg 1). But fghg 1 j h 2 Hg = H for each g 2 H. So g(b( h2h J h )) = B( h2h J h ) for each g 2 H by Lemma 3.1. (2) By Theorem 2 in [6], J h J h 1 = e h C for some idempotent e h in C. So by Lemma 3.1, B( h2h J h ) = B( h2h J 2 h ) = B( h2hj h J h 1) = B( h2h e h ). Noting h2h e h by e, we have B( h2h J h ) = Be where e is an idempotent in C. (3) By (1) B( h2h J h ) is invariant under H and by (2) B( h2h J h ) = Be, so for h 2 H, h(e) = e (for e is the identity of Be). Since B is a Galois algebra over R with Galois group G, there exists a G-Galois system fa i ; b i in B, i = 1; 2; :::; mg for some integer m such that P m i=1 a ig(b i ) = 1;g for each g 2 G. Hence e P m i=1 a ih(b i ) = e 1;h ; and P m i=1 (ea i)h(eb i ) = e 1;h for each h 2 H. Therefore, e = P m i=1 (ea i)(h(eb i ) (eb i )) for each h 6= 1 in H. Now by hypothesis, h2h J h 6= f0g, so e 6= 0. This implies that hj Be 6= 1 whenever h 6= 1 in H. Thus, Hj Be = H. LEMMA 3.3. Let B be a Galois algebra over R with Galois group G, H a subgroup of G, e the same as in Lemma 3.2, and Jh 0 = fb 2 Be j bx = h(x)b for all x 2 Beg for each h 2 H = Hj Be. Then, J 0 h = ej h for each h 2 H = Hj Be. Proof. It is clear that ej h Jh 0. Conversely, for any b 2 J h 0, b = eb and bx = h(x)b for each x 2 Be. Hence for any y 2 B, by = (eb)y = b(ye) = h(ye)b = h(y)eb = h(y)b. Therefore, b 2 J h. So, b = eb 2 ej h. Thus, J 0 h = ej h. Next are two \local" structure theorems for B( h2h J h ) where H is a subgroup of G such that h2h J h 6= f0g. 4

Theorem 3.4. Let B be a Galois algebra over R with Galois group G. If H is a subgroup of G such that h2h J h 6= f0g, then B( h2h J h )(= Be) is a Galois H-separable extension with Galois group Hj Be = H. Proof. By Lemma 3.2-(2), B( h2h J h ) = Be for some idempotent e in C, and by the proof of Lemma 3.2-(1) and (3), P m i=1 (ea i)h(eb i ) = e 1;h for each h 2 H = Hj Be. Hence Be is a Galois extension with Galois group Hj Be = H. Moreover, by Theorem 2 in [6], J h J h 1 = e h C for some idempotent e h in C and e h e = e by the proof of Lemma 3.2. Therefore, by Lemma 3.3, J 0 h J 0 h 1 = (ej h )(ej h 1) = ej h J h 1 = ee h C = ec which is the center of Be. Thus, Be is a Galois H-separable extension with Galois group Hj Be = H ([7, Theorem 2]). With the maximality property of the subset H of G such that h2h J h 6= f0g, we show that H is indeed a subgroup of G and that B( h2h J h ) becomes a central Galois algebra, a stronger Galois extension than the Galois H-separable extension as given in Theorem 3.4. LEMMA 3.5. Let B be a Galois algebra over R with Galois group G and H a maximal subset of G such that h2h J h 6= f0g. Then H is a subgroup of G. Proof. For any g; h 2 H, we claim that gh 2 H. In fact, suppose that gh 62 H. Then ( h2h J h )J gh = f0g by the maximality property of H. Hence B( h2h J h )J gh = f0g. Since g; h 2 H and BJh 2 = BJ h for all h 2 H by Lemma 3.1, f0g = B( h2h J h )J gh = B( h2h J h )(J g J h )J gh = B( h2h J h )(I g J gh )J gh ([6; Proposition 2]) = B( h2h J h )(I g J gh ) = B( h2h J h )(J g J h ) = B( h2h J h ) 6= f0g: This is a contradiction. Thus, gh 2 H. But then H is a subgroup of G for G is nite. 5

Theorem 3.6. Let B be a Galois algebra over R with Galois group G. If H is a maximal subset of G such that h2h J h 6= f0g, then H is a subgroup of G and B( h2h J h )(= Be) is a central Galois algebra with Galois group Hj Be = H. Proof. By Lemma 3.5, H is a subgroup of G. Since B is a Galois algebra over R with Galois group G, B = Pg2G J g. Hence Be = Pg2G ej g = ( Ph2H ej h) ( Pg62H ej g). by Lemma 3.3, Jh 0 = ej h for each h 2 H = Hj Be. By the maximality property of H, BeJ g = B( h2h J h )J g = f0g for each g 62 H. Hence ej g = f0g for each g 62 H. Thus, Be = Ph2H J h 0. Also, J h 0 J h 0 1 = (ej h )(ej h 1) = ej h J h 1 = ec which is the center of Be. Moreover, B is a Galois R-algebra, so it is a separable R-algebra. Thus, Be is a separable algebra over Re ([2, Proposition 1.11, page 46]). Therefore, Be is a central Galois algebra over Ce ([4, Theorem 1]). To obtain a structure theorem for a Galois algebra, we need a lemma. LEMMA 3.7. Let B be a Galois algebra over R with Galois group G and e a nonzero idempotent in C G. Then Be is a Galois algebra over Re with Galois group Gj Be = G. Proof. Since e 2 C G, Be is invariant under G. By hypothesis, B is a Galois algebra over R with Galois group G, so there exists a G-Galois system for B fa i ; b i in B, i = 1; 2; :::; mg for some integer m such that P m i=1 a ig(b i ) = 1;g for each g 2 G. Hence e P m i=1 a ig(b i ) = e 1;g ; and P m i=1 (a ie)g(b i e) = e 1;g for each g 2 G. Therefore, fa i e; b i e in Be, i = 1; 2; :::; mg is a G-Galois system for Be and e = P m i=1 (a ie)(g(b i e) b i e) for each g 6= 1 in G. But e 6= 0, so gj Be 6= 1 whenever g 6= 1 in G. Thus, Be is a Galois algebra over Re with Galois group Gj Be = G. THEOREM 3.8. Let B be a Galois algebra over R with Galois group G. Then there are orthogonal idempotents fe i j i = 1; 2; :::; m for some integer mg in C and subgroups H i 6

of G such that Be i is a central Galois algebra with Galois group H i for each i = 1; 2; :::; m P and B = Pm i=1 Be i or B = ( Pm i=1 Be m i) Ce where e = 1 i=1 e i and Ce = Be is a commutative Galois algebra with Galois group Gj Ce = G. Proof. Let H 1 be a maximal subset of G such that h2h1 J h 6= f0g. Then H 1 is a subgroup of G by Lemma 3.5. By Theorem 3.6, there exists an idempotent e 1 in C such that B( h2h1 J h )(= Be 1 ) is a central Galois algebra over Ce 1 with Galois group H 1 j Be1 = H1. Let H 1, H 2 = g 2 H 1 g 1,..., 2 H k = g k H 1 g 1 k be all the distinct conjugates of H 1 in G for some g i 2 G (if H 1 is a normal subgroup of G, k = 1 and g 1 = 1). Since for each i = 2; 3; :::; k, hi 2H i J hi = h2h1 J gi hg 1 i = g i ( h2h1 J h ), H i is also a maximal subset (subgroup) of G such that hi 2H i J hi 6= f0g by the maximality property of H 1. Hence, by Theorem 3.6 again, for each i = 2; 3; :::; k, there exists an idempotent e i in C such that B( hi 2H i J hi )(= Be i ) is a central Galois algebra over Ce i with Galois group H i j Bei = Hi. Moreover, (Be i )(Be j ) = B( hi 2H i J hi )B( hj 2H j J hj ) = Be i ij by Lemma 3.1 and the maximality property of H i and H j ; and so e 1, e 2,..., e k are orthogonal. Thus, B = ( Pk B(1 i=1 e i). In case that i=1 e i = 1, we have B = Pk case that i=1 e i 6= 1, we have 1 i=1 Be i) i=1 Be i, and so we are done. In i=1 e i 6= 0. Since fh 1 ; H 2 ; :::; H k g = fg i H 1 gi 1 j i = 1; 2; :::kg are all the distinct conjugates of H 1 in G, fgh 1 g 1 ; gh 2 g 1 ; :::; gh k g 1 g = fh 1 ; H 2 ; :::; H k g for any g 2 G. Hence, that g(b( hi 2H i J hi )) = B( hi 2H i J ghi g 1) implies that fg(e 1 ); g(e 2 ); :::; g(e k )g = fe 1 ; e 2 ; :::; e k g for all g 2 G. Hence, g(1 i=1 e i) = 1 i=1 e i for all g 2 G, and so 1 i=1 e i a nonzero idempotent in C G. Thus, by Lemma 3.7 B(1 i=1 e i) is a Galois algebra over R(1 i=1 e i) with Galois group Gj B(1 i=1 e i) = G. Let e = 1 i=1 e i. Then Be is a Galois algebra over Re with Galois group Gj Be = G. In case that ejg = f0g for each g 6= 1 in G, we have that Be = Pg2G ej g = ej 1 = ec. Thus, Be is a commutative Galois algebra with Galois group Gj Be = G, and so we are done. In case that ejg 6= f0g for some g 6= 1 in G, we can obtain a subgroup H k+1 and an idempotent e k+1 in C such that Be k+1 is a central Galois algebra with Galois group H k+1 j Bek+1 = Hk+1. Let E = fe g j g 2 G and J g J g 1 = e g Cg. 7

By the proof of Lemma 3.2-(2), each e i is contained in the Boolean algebra generated by the elements in E which is nite. Consequently, in nite steps, we have orthogonal idempotents e i in C and subgroups H i, i = 1; 2; :::; m for some integer m such that Be i is a central Galois algebra with Galois group H i for each i = 1; 2; :::; m and B = Pm i=1 Be i P or B = ( Pm i=1 Be m i) Ce where e = 1 i=1 e i and Ce is a commutative Galois algebra with Galois group Gj Ce = G. This completes the proof. If we further assume in Theorem 3.8 that J g 6= f0g for each g 6= 1 in G, then we can show that G = [ m i=1 H i. THEOREM 3.9. Let B be a Galois algebra over R with Galois group G and J g 6= f0g for each g 6= 1 in G. Then there are orthogonal idempotents fe i j i = 1; 2; :::; m for some integer mg in C and subgroups H i of G such that G = [ m i=1 H i, Be i is a central Galois algebra with Galois group H i B = ( Pm i=1 Be i) Ce where e = 1 with Galois group Gj Ce = G. for each i = 1; 2; :::; m, and B = Pm i=1 Be i or P m i=1 e i and Ce is a commutative Galois algebra Proof. By Theorem 3.8, there are orthogonal idempotents fe i j i = 1; 2; :::; m for some integer mg in C and subgroup H i of G such that Be i is a central Galois algebra over Ce i with Galois group H i j Bei = Hi for each i = 1; 2; :::; m and B = Pm i=1 Be i P or B = ( Pm i=1 Be m i) Ce where e = 1 i=1 e i and Ce is a commutative Galois algebra with Galois group Gj Ce = G. Moreover, Hi 's are maximal subsets (subgroups) of G such that h2hi J h 6= f0g. In case that B = Pm i=1 Be i, for any g 6= 1 in G, f0g 6= BJ g = Pm i=1 Be ij g = Pm B( i=1 h i 2H i J hi )J g. Hence, there is some H i such that ( hi 2H i J hi )J g 6= f0g. Therefore, g is contained in some H i by the maximality property of H i. Thus, G = [ m i=1 H i. In case that B = Pm i=1 Be i Ce, we have ej 1 = Pg2G ej g. Hence, ej g = f0g for any g 6= 1 in G. Therefore, for any g 6= 1 in G, 8

f0g 6= BJ g = ( Pm i=1 Be ij g ) CeJ g = Pm i=1 Be ij g = Pm B( i=1 h i 2H i J hi )J g. Thus, similar to the above argument, G = [ m i=1 H i. This completes the proof. In [3], let f: G G! U(R) be a factor set from G G to the P set of units of R, that is, f(g; h)f(gh; k) = f(h; k)f(g; hk) for all g, h, and k in G. RG f = g2g RU g is called a projective group algebra over R if RG f is an algebra with a free basis fu g j g 2 Gg over R where the multiplications are given by (r g U g )(r h U h ) = r g r h U g U h and U g U h = f(g; h)u gh for r g ; r h 2 B and g; h 2 G. It was shown that if B is a central Galois algebra over C with inner Galois group G, then B is an Azumaya projective group algebra over C ([3, Theorem 6]). We next generalize the above structure theorem for a central Galois algebra with an inner Galois group given by F. R. DeMeyer. THEOREM 3.10. If B is a Galois algebra over R with an inner Galois group G, then B is either a direct sum of Azumaya projective group algebras or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. Proof. By Theorem 6 in [3], a central Galois algebra with an inner Galois group is an Azumaya projective group algebra, so this is a consequence of Theorem 3.8. We conclude the present paper with two examples of a Galois algebra B which is not a central Galois algebra with Galois group G such that J g 6= f0g for some g 2 G and J h = f0g for some h 2 G, and B = Pm i=1 Be i or B = ( Pm i=1 Be i) Ce as given in Theorem 3.8. EXAMPLE 1. Let R[i; j; k] be the real quaternion algebra over R, B = R[i; j; k] R[i; j; k], and G = f1; g i ; g j ; g k ; g; gg i ; gg j ; gg k g where g i (a 1 ; a 2 ) = (ia 1 i 1 ; ia 2 i 1 ), g j (a 1 ; a 2 ) = (ja 1 j 1 ; ja 2 j 1 ), g k (a 1 ; a 2 ) = (ka 1 k 1 ; ka 2 k 1 ), and g(a 1 ; a 2 ) = (a 2 ; a 1 ) for all (a 1 ; a 2 ) in B. Then, 9

(1) B is a Galois extension with a G-Galois system: fa 1 = (1; 0); a 2 = (i; 0); a 3 = (j; 0); a 4 = (k; 0); a 5 = (0; 1); a 6 = (0; i); a 7 = (0; j); a 8 = (0; k); b 1 = 1 4 (1; 0); b 2 = 1 (i; 0); b 1 4 3 = (j; 0); b 1 4 4 = (k; 0); b 4 5 = 1(0; 1); b 1 4 6 = (0; i); b 1 4 7 = (0; j); b 4 8 = 1(0; k)g. 4 (2) B G = f(r; r) j r 2 Rg = R. (3) By (1) and (2) B is a Galois algebra over R with Galois group G. (4) C = R R. (5) By (3) and (4) B is not a central Galois algebra with Galois group G. (6) J 1 = C = R R, J gi = (Ri) (Ri), J gj = (Rj) (Rj), J gk = (Rk) (Rk), and J g = J ggi = J ggj = J ggk = f0g. (7) H 1 = f1; g i ; g j ; g k g is a maximal subset (subgroup) of G such that h2h1 J h 6= f0g, and B( h2h1 J h )(= Be 1 = B) is a central Galois algebra over C with Galois group H 1 j Be1 = H1, and so B = Pm i=1 Be i where m = 1 and Be i is a central Galois algebra over Ce i with Galois group H i j Bei = Hi for each i. But [ m i=1 H i = H 1 6= G. EXAMPLE 2. Let R[i; j; k] be the real quaternion algebra over R, D the eld of complex numbers, B = R[i; j; k] (D R D), and G = f1; g i ; g j ; g k g where g i (a; d 1 d 2 ) = (iai 1 ; d 1 d 2 ), g j (a; d 1 d 2 ) = (jaj 1 ; d 1 d 2 ), and g k (a; d 1 d 2 ) = (kak 1 ; d 1 d 2 ) for all (a; d 1 d 2 ) in B, where d is the conjugate of the complex number d. Then, (1) B is a Galois extension with a G-Galois system: fa 1 = (1; 0); a 2 = (i; 0); a 3 = (j; 0); a 4 = (k; 0); a 5 = (0; 1 1); a 6 = (0; p 1 1); a 7 = (0; 1 p 1); a 8 = (0; p 1 p 1); b1 = 1(1; 0); b 1 4 2 = (i; 0); b 1 4 3 = (j; 0); b 1 4 4 = (k; 0); b 4 5 = 1(0; 1 1); b 4 6 = 1 (0; p 1 1 1); b 4 7 = (0; 1 p 1); b 4 8 = 1(0; p 1 p 1)g. 4 (2) B G = R (R R) = R R. (3) By (1) and (2) B is a Galois algebra over R R with Galois group G. (4) C = R (D R D). 10

(5) By (3) and (4) B is not a central Galois algebra with Galois group G. (6) J 1 = C = R (D R D), J gi = R(i; 0), J gj = R(j; 0), J gk = R(k; 0). (7) H 1 = f1; g i ; g j ; g k g = G is a maximal subset (subgroup) of G such that h2h1 J h 6= f0g, and B( h2h1 J h )(= B(1; 0) = R[i; j; k]) is a central Galois algebra over C(1; 0)(= R) with Galois group H 1 j B(1;0) = H1. Let e 1 = (1; 0) and e = (0; 1 1). Then B = ( Pm i=1 Be i) Ce where m = 1, Be i is a central Galois algebra over Ce i with Galois group H i j Bei = Hi for each i = 1; 2; :::; m and Ce = Be is a commutative Galois algebra with Galois group Gj Ce = G. REFERENCES [1] R. Alfaro and G. Szeto, On Galois Extensions of an Azumaya Algebra, Comm. in Algebra, 25(6)(1997) 1873-1882. [2] F.R. DeMeyer and E. Ingraham, \Separable algebras over commutative rings", Volume 181, Springer Verlag, Berlin, Heidelberg, New York, 1971. [3] F.R. DeMeyer, Some Notes on the General Galois Theory of Rings, Osaka J. Math., 2(1965) 117-127. [4] M. Harada, Supplementary Results on Galois Extension, Osaka J. Math., 2(1965), 343-350. [5] S. Ikehata and G. Szeto: On H-skew polynomial rings and Galois extensions, \Rings, Extension and Cohomology" 113-121, Lecture Notes in Pure and Appl. Math., 159, Dekker, New York, 1994. [6] T. Kanzaki, On Galois Algebra Over A Commutative Ring, Osaka J. Math., 2(1965), 309-317. [7] K. Sugano, On a Special Type of Galois Extensions, Hokkaido J. Math., 9(1980) 123-128. 11

[8] G. Szeto and L. Ma, On center Galois extensions over rings, Glasnik Matematicki, 24(1989), 11-16. [9] G. Szeto and L. Xue, On Three types of Galois Extensions of rings, Southeast Asian Bulletin of Mathematics, 23(1999) 731-736. [10] G. Szeto and L. Xue, On Characterizations of a Center Galois Extension, International Journal of Mathematics and Mathematical Sciences, Vol. 23, No. 11(2000) 753-758. 12