When is the Ring of 2x2 Matrices over a Ring Galois?
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1 International Journal of Algebra, Vol. 7, 2013, no. 9, HIKARI Ltd, When is the Ring of 2x2 Matrices over a Ring Galois? Audrey Nelson Department of Mathematics Bradley University, Peoria, Illinois USA anelson@mail.bradley.edu George Szeto Department of Mathematics Bradley University, Peoria, Illinois USA szeto@fsmail.bradley.edu Copyright c 2013 Audrey Nelson and George Szeto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let R be a ring with 1, M 2 (R) the ring of 2 2 matrices over R, and S the Sugano quaternion ring over R. Then, S is isomorphic with a subring of M 2 (R) if and only if 2 is not a zero divisor in R, and S = M 2 (R) if and only if 2 is invertible in R. Moreover, if 2 is invertible in R, then M 2 (R) is a Galois extension of R with an abelian inner Galois group G of order 4. This implies that the rings of 2 2 matrices over the real field and complex field are central Galois algebras induced by the central Galois algebra of 2 2 matrices over the rational field with Galois group G. Mathematics Subject Classification: 16S35, 16W20 Keywords: Sugano quaternion rings, Matrix rings, Galois extensions, Abelian inner Galois groups
2 440 Audrey Nelson and George Szeto 1 Introduction Let R be the field of real numbers, and A the real quaternion ring as given in [5 with free generators, {1,i,j,k} over R such that i 2 = j 2 = k 2 = 1,ij = k, jk = i, ki = j, ji = k, kj = i and ik = j. It is well known that A is a central division algebra over R isomorphic with a subring of the ring of 2 2 matrices over the field of complex numbers. The real quaternion ring A has been widely used in the study of non-commutative rings. The purpose of the present paper is to study another kind of quaternion ring, the Sugano quaternion ring over a ring with 1. A ring S is called a Sugano quaternion ring with free generators {1,I,J,K} over a ring R such that for each r R, r1 =1r, ri = Ir,rJ = Jr,rK = Kr,I 2 = 1,J 2 = K 2 =1,IJ = K, JI = K, IK = J, KI = J, JK = I,KJ = I. We shall show that S is isomorphic with a subring of M 2 (R) if and only if 2 is not a zero divisor in R, and that S = M 2 (R) if and only if 2 is invertible in R. Moreover, observing that {1,I,J,K} are invertible in S, we have an inner automorphism group G of S induced by {1,I,J,K}: G = {g 1,g I,g J,g K g 1 (x) =x, g I (x) = IxI 1,g J (x) =JxJ 1,g K (x) =KxK 1 } for each x S. Then G is an abelian group of order 4 and S G = R where S G is the subring of the elements fixed under each element in G. We shall show that if 2 is invertible in R, then S is a Galois extension of S G with Galois group G. In particular, S is a central Galois algebra over R if and only if R is a commutative ring with 2 invertible in R. Thus if 2 is invertible in R, then M 2 (R) is a Galois extension of R with Galois group G; and M 2 (R) is a central Galois algebra over R if and only if R is a commutative ring with 2 invertible in R. When R is either the real or complex field, M 2 (R) is a central Galois algebra over R induced by the central Galois algebra M 2 (Q) over the rational field Q with an isomorphism Galois group G. 2 Preliminary Throughout the paper, let R be a ring with 1, M 2 (R) the ring of 2 2 matrices over R, and S =[1,I,J,K the Sugano quaternion ring over R as defined in Section 1. As given in [1, 2, 3, 5, let A be a ring with 1, G a finite automorphism group of A, and A G the subring of the elements fixed under each element in G. If there exist {x i,y i i =1,,n, i x ig(y i )=δ 1g } for some integer n, then A is call a Galois extension of A G with Galois group G, and {x i,y i i =1,,n} is called a G- Galois system for A. In particular, if A G C where C is the center of A, then the Galois extension A is called a Galois algebra, and if A G = C, then A is called the central Galois algebra.
3 When is the ring of 2x2 matrices over a ring Galois? The Sugano Quaternion Ring In this section, keeping the definitions and notations in Section 2, we shall show that the Sugano quaternion ring S over a ring R is isomorphic with a subring of M 2 (R) if and only if 2 is not a zero divisor in R. We begin with some properties of S. Lemma 3.1 Let C be the center of R. If2 is not a zero divisor in R, then C is the center of S. Proof. Let Z be the center of S. Then for an x S, x = a + bi + cj + dk for some a, b, c, d R such that rx = xr for each r R. This implies that a, b, c, d are in C. Also, xi = Ix,soaI b + ck dj = ai b ck + dj; and so c =0 and d = 0. Moreover, xj = Jx,sob =0. Thusx = a. Therefore Z C. Itis clear that C Z. We have Z = C. Let E = [ [ 0 1,F= 1 0 [ 1 0,P= 0 1 [ 0 1,T= 1 0. We define a map α : S M 2 (R) by extending α(1) = E,α(I) =F, α(j) = P, α(k) =T linearly to S; that is, α(a + bi + cj + dk) =ae + bf + cp + dt where {a, b, c, d} are in R. Lemma 3.2 Let α : S M 2 (R) be given above. Then α is a ring homomorphism. Proof. It is clear that α preserves addition. Next we show that α preserves the multiplication of the basis elements {1,I,J,K}. That is, α(ii) = α( 1) = E = FF = α(i)α(i),α(ij)=α( K) = T = FP = α(i)α(j), and similarly for other elements. Then we can see that α preserves the multiplication by extending the above results linearly to S. Theorem 3.3 Let α : S M 2 (R) be given in Lemma 3.2. Then, α is one-to-one if and only if 2 is not a zero divisor in R. Proof. By Lemma 3.2,α is a ring homomorphism, so α is one-to-one if and only if the kernel of α is {0}. Let x = a + bi + cj + dk ker(α), Then α(x) is the 0-matrix in M 2 (R); that is, ae+bf+cp+dt is the 0-matrix. This imples a + c =0,b+ d =0, b + d =0,a c = 0. Hence 2a =0, 2d =0. Thus ker(α) ={0} if and only if 2 is not a zero divisor in R. Next we show an equivalent condition for the Sugano quaternion ring S isomorphic with M 2 (R).
4 442 Audrey Nelson and George Szeto Theorem 3.4 Let α be given in T heorem 3.3. Then, S = M 2 (R) by α if and only if 2 is invertible in R. Proof. By T heorem 3.3, it suffices to show that α : S M 2 (R) isontoif and only if 2 is invertible in R. Let x S such that x = a + bi + cj + dk for some a, b, c, d R. [ 1 0 α(x) =. Then a + c =1,b+ d =0, b + d =0,a c = 0. We have 2a =1, 2d =0. Thus x = J S if and only if 2 is invertible in R. Similarly, 2 is invertible in R if and only if there exist y, z, w S such that α(y) = [ 0 1,α(z) = [ 1 0,α(w) = [ 0 1 Noting that {α(x),α(y),α(z),α(w)} is an R-basis for M 2 (R) and α is R-linear, we conclude that α is onto if and only if 2 is invertible in R. Corollary 3.5 Let R be a field. Then, S = M 2 (R) if and only if the characteristic of R is not 2. Proof. This is immediate by T heorem The Galois Matrix Ring Keeping the notations and definitions as given in Section 2, let S = R + RI + RJ + RK be the Sugano quaternion ring over a ring R and M 2 (R) the ring of 2 2 matrices over R. Observing I,J, and K are invertible in S, we have an inner automorphism group G of S induced by {1,I,J,K}. Denote G by {g 1,g I,g J,g K }. Then G is an abelian group of order 4. We shall show that if 2 is invertible in R, then S is a Galois extension of R with Galois group G; and so M 2 (R) is also a Galois extension of R with an abelian Galois group of order 4. We begin with some properties of G. Lemma 4.1 Let G be given above. Then G is an abelian group of order 4. Proof. It is straightforward to show that g 2 I = g 2 J = g2 K = g 1,g I g J = g J g I = g K,g I g K = g K g I = g J,g K g J = g J g K = g I.
5 When is the ring of 2x2 matrices over a ring Galois? 443 Lemma 4.2 Let S G = {x S g(x) =x} for each g G. Then S G = R. Proof. It is clear that R S G. Conversely, let x = a + bi + cj + dk S G. Then xi = Ix implies that ai b + ck dj = ai b ck + dj. Hence c = c, d = d. Since 2 is not a zero divisor by hypothesis, c = d =0. Similarly, Jx = xj implies that b =0. Thusx = a R; and so R G R. Lemma 4.3 If 2 is invertible in R, then S is a Galois extension of R with an abelian inner Galois group G of order 4 as given in Lemma 3.1. Proof. Let {x 1 =1,x 2 = I,x 3 = J, x 4 = K; y 1 =1/4,y 2 = I/4,y 3 = J/4,y 4 = K/4}. Then we claim that {x i,y i } is a G-Galois system for S over R. In fact, x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 =1/4 I 2 /4+J 2 /4+K 2 /4= 1/4 ( 1)/4+1/4+1/4 = 1, and x 1 g I (y 1 )+x 2 g I (y 2 )+x 3 g I (y 3 )+x 4 g I (y 4 )= 1/4 +II( I/4)( I)+JI(J/4)( I)+KI(K/4)( I) = 1/4 +1/4 1/4 1/4 = 0; x 1 g J (y 1 )+x 2 g J (y 2 )+x 3 g J (y 3 )+x 4 g J (y 4 )=1/4+IJ( I/4)J + JJ(J/4)J + KJ(K/4)J =1/4 1/4+1/4 1/4 = 0, and x 1 g K (y 1 )+x 2 g K (y 2 )+x 3 g K (y 3 )+ x 4 g K (y 4 )=1/4+IK( I/4)K +JK(J/4)K +KK(K/4)K =1/4 1/4 1/4+ 1/4 = 0. Thus S is a Galois extension of S G with abelian inner Galois group G by Lemma 3.1. By Lemma 3.2,S G = R, so the proof is complete. Next we show a sufficient condition for a Galois M 2 (R) with an abelian inner Galois group. Theorem 4.4 Let α : S M 2 (R) be given in Lemma 3.2. If2 is invertible in R, then M 2 (R) is a Galois extension of R with an abelian inner Galois group induced by {α(1),α(i),α(j),α(k)}. Proof. Since 2 is invertible in R by hypothesis, S is a Galois extension of R with an abelian inner Galois group G by Lemma 4.3 and S = M 2 (R) by T heorem 3.4. Thus M 2 (R) is a Galois extension of R with an abelian inner Galois group induced by {α(1),α(i),α(j),α(k)}. In case R is commutative, we obtain an equivalent condition for a central Galois algebra M 2 (R). Theorem 4.5 Let R be a commutative ring with 1. Then, M 2 (R) is a central Galois algebra over R with an belian inner Galois group of order 4 if and only 2 is invertible in R. Proof. The sufficiency is by T heorem 4.4. Conversely, since M 2 (R) isa central Galois algebra over R with an abelian inner Galois group of order 4, 4 is invertible in R ([4, Proposition 4); and so 2 is invertible in R.
6 444 Audrey Nelson and George Szeto Corollary 4.6 Let Q, R, C, be the rational, real and complex field respectively. Then M 2 (R) and M 2 (C) are central Galois algebras induced by the Galois algebra M 2 (Q) with an abelian inner Galois group of order 4. Proof. Since 2 is invertible in Q,M 2 (Q) is a central Galois algebra contained in M 2 (R) and M 2 (C), the corollary is immediate by T heorem 4.5. References [1 F.R. DeMeyer, Galois Theory in Separable Algebras over Commutative Rings, Illinois J. Math., 10, 1966, [2 X.L. Jiang and G. Szeto, On the Injective Galois Map, International J. Algebra, 5, Vol 6, 2012, [3 T. Kanzaki, On Galois Algebra over a Commutative Ring, Osaka J. Math., 2, 1965, [4 K. Sugano, On a Special Type of Galois Extensions, Hokkaido J. Math., 9, 1980, [5 G. Szeto, On Azumaya Crossed Products, Bull. Malaysian Math. Sci. Soc., 2, Vol 25 (2A), 2012, Received: April 23, 2013
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