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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 3, December 1974 A CHARACTERIZATION OF NONCOMMUTATIVE QUATERNIONRINGS CARL W. KOHLS ABSTRACT. Noncommutative quaternion subrings of the ring of 4x4 matrices over a formally real field are characterized in terms of the form of commutators and the multiplicity of characteristic roots. 1. Introduction. From the proof of Lemma 10 in [2], one can easily deduce the following result: A subring T of the ring of all 2x2 matrices over a formally real field F is similar to a ring of matrices over the real closure of F having the form [ a, ] if and only if every matrix in T with a character- istic root in the real closure of F is a scalar matrix. W. H. Reynolds sug- gested to the author that an analogous result leading to isomorphisms between rings of 4 x 4 matrices over F and rings of quaternions would be worth seek- ing. We present such a theorem here for noncommutative rings. As might be expected, the situation is more involved than in the 2x2 case, so the characterization we obtain is somewhat different. In particular, the similar- ity transformation disappears. It is a pleasure to thank the referee for his excellent suggestions, which led to concise proofs of Lemmas 1 and 3. 2. Preliminaries. The ring of all A x A matrices over an arbitrary for- mally real field F will be denoted by F . We write [Aß] for the commutator AB - BA oí matrices A and B in F . The division ring Q of quaternions over F will be viewed as the set of elements in F . having the form 7 j"l abed —b a —d c —c d a —b -d -c b a and the opposite ring g as the set of elements in F . having the form Received by the editors October 1, 1973. AMS iMOS) subject classifications (1970). Primary 15A30, 16A42; Secondary 15A18, 15A57, 16A40. Copyright © 1974, American Mathematical Society License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 351 352 C. W. KOHLS abed -b a d -c -c -d a b rd c -b a Note that gop is orthogonally similar to Q, through conjugation with the permutation matrix that interchanges the second and fourth columns. Also Q commutes elementwise with Qop, and Q O gop is the ring of scalar matrices. Let S denote the seven-dimensional subspace of F . consisting of all matrices that are the sum of a scalar matrix and a skew matrix. Clearly g u Qop Ç S. By considering dimensions, one sees immediately that S is the subspace of F generated by Q U Q"^. We shall be concerned with the following conditions on a subset T of 4 (1) Each commutator of matrices in T is skew. (2) No matrix in T has a simple characteristic root. (3) There are two skew matrices in T O Q or in T O Qv with nonzero commutator. From the definition of S, each commutator of elements in S is equal to a commutator of skew matrices, which is also skew. Thus, S satisfies condi- tion (1). It is easy to construct examples to show that S does not satisfy condition (2), and that some subsets of F disjoint from S do satisfy it. However, by direct calculation one finds that the characteristic roots of the two matrices in Q U Qop with first row {abed) ate a ± (b + c + d ) y/— 1, a ± {b2 + c2 + d2)1/2yf- 1. Hence Q U Qop satisfies condition (2). 3. The main result. We first prove three lemmas, from which the char- acterization follows rather quickly. Lemma 1. // T is a subset of F . that satisfies conditions (1) and (3), then TÇS. Proof. Choose skew matrices Aj and A. in T O Q or in T n Qop with [A.A ] / 0. Let B £ T, and write B = M + K, where M is symmetric and K is skew. For / = 1 and 2, [MA .] is symmetric, [KA .] is skew, and [MA .] + [KA .] = [BA .] is skew; hence [MA .] = 0. Let r,, r , r and r be the char- acteristic roots of M in the real closure of F, and select a nonzero vector X. in the characteristic subspace V. corresponding tor. (¿=1,2, 3, A). Since [MA,]= [MA 1= 0, \X., A X., A X.} C V.. If c.X. + c.A.X. + 1 2 ' i 1 i 2 i — i 07 111 c.A.X.= 0, where c c, and c„ are in the real closure of F, then c.A, + 2 2 i 0' 1 2 11 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVEQUATERNION RINGS 353 c2A2 is skew with characteristic root - cQ, and is either invertible or zero, so cQ = 0 and c A + c A = 0; because [A A ] 4 0, this implies that c = c„ = 0. Hence the dimension of V ■ is at least three. Thus V. n V. , 4\0] 2 l 7 2+1 (z = 1, 2, 3), whence r. - r = r = r . It follows that M is a scalar matrix, and B £ S. Lemma 2. If B £ S and\ß\ satisfies condition (2), then B £ Q U Qop. Proof. Let B = [b..]. The characteristic polynomial of B, which is * ♦(£**)Zb2j){bll-x)2 + {b12bi4-blib24 + bl4b25)2, must be the square of a quadratic polynomial in b. — x. It follows quickly that 2>¿ = ±Xbl2bi4-bnbu + bub2i). 7<7 The choice of a minus sign i leads to {bl2 + bi4)2 + {bn -b24)2 + {b14 + b2i)2 = 0, so that &12 = ~¿34' è13 = ¿24' è14 = -&23' and B £ Q. Similarly, the choice of a plus sign yields B £ gop'. Thus, B £ Q UQ09. Lemma 3. // T is a subset of Q U 2°p that is closed under multiplica- tion, then TÇQ or TÇeop. Proof. Suppose T t Qop, and choose B £ T with B '4 0e*. Then [Aß] 4 0 for some A £ Q. 1Í C £ T D Qop, either C = 0 or [A{BC)]= [Aß]C (¿0. Since BC £ Q U g^, we have BCeÇ, whence C e Q. Thus TÇQ. Theorem. A noncommutative subring T of F , is a subring of either Q or Q0^ if and only if it satisfies conditions (1) and (2). Proof. We observed in §2 that S satisfies condition (1), and that Q U 2°P satisfies condition (2). Conversely, assume that T satisfies conditions (1) and (2). Choose A and B in T with C =.[Aß]4 0. Then C is skew, so by Lemma 2, C £ Q U Q"9. This implies that C2 is a scalar matrix. If [CA]= 0, then C2 = C[Aß] = [A(CB)] is skew, so that C2 = 0, whence C = 0, a contradiction. Thus D = [CA]4 0 and D is skew. By a similar argument D e Q U Qop and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 354 C. W. KOHLS [DC] 4 0. It follows that \C,D\ Ç TO Q or \C, D\ Ç Tn Qop. Therefore T satisfies condition (3). Lemmas 1, 2 and 3 together now show that T is a subring of either Q or Q09. Remark 1. The observation that C is a commutator is contained in the proof of Theorem 18 in [1, Chapter l]. Actually, the nilpotency of C is an immediate consequence of that theorem, but we prefer to give a self-contained proof. Remark 2. Obviously conditions (1) and (2) are also necessary for com- mutative subrings. But it is easy to see that they are not sufficient. For example, there are rings of diagonal matrices that satisfy them but are not even contained in S. REFERENCES 1. Irving Kaplansky, Lie algebras and locally compact groups, Univ. of Chicago Press, Chicago, 111., 1971. MR 43 #2145. 2. C. W. Kohls and W. H« Reynolds, Embedding rings with a maximal cone and rings with an involution in quaternion algebras, Trans. Amer. Math. Soc. 176 (1973), 411-419. MR 47 #1857. DEPARTMENT OF MATHEMATICS, SYRACUSE UNIVERSITY, SYRACUSE, NEW YORK 13210 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use.
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