Jordan isomorphisms of triangular matrix algebras over a connected commutative ring

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1 Linear Algebra and its Applications 312 (2000) Jordan isomorphisms of triangular matrix algebras over a connected commutative ring K.I. Beidar a,,m.brešar b,1, M.A. Chebotar c a Department of Mathematics, National Cheng-Kung University, Tainan, Taiwan, ROC b Department of Mathematics, University of Maribor, Maribor, Slovenia c Department of Mechanics and Mathematics, Moscow State University, Moscow, Russian Federation Received 17 December 1999; accepted 24 February 2000 Submitted by C.-K. Li Abstract Let C be a 2-torsionfree commutative ring with identity 1, and let T r (C), r 2, be the algebra of all upper triangular r r (r 2) matrices over C.ThenC contains no idempotents except 0 and 1 if and only if every Jordan isomorphism of T r (C) onto an arbitrary algebra over C is either an isomorphism or an anti-isomorphism Published by Elsevier Science Inc. All rights reserved. Keywords: Triangular matrix algebra; Jordan isomorphism Let C be a commutative ring with identity 1 and let T and A be algebras over C. Recall that a bijective C-linear map ϕ : T A is called a Jordan isomorphism if ϕ(ab + BA) = ϕ(a)ϕ(b) + ϕ(b)ϕ(a) for all A, B T. Isomorphisms and antiisomorphisms are obvious examples of Jordan isomorphisms, and often it turns out that they are actually the only possible examples. However, not always: Let ε C be an idempotent and let A Ã be an anti-automorphism of the algebra T.ThenA εa + (1 ε)ã is a Jordan automorphism of T which is neither an automorphism nor an anti-automorphism, unless one of the ideals (1 ε)t or εt is commutative. Corresponding author. addresses: beidar@mail.ncku.edu.tw (K.I. Beidar), bresar@uni-mb.si (M. Brešar), mchebotar@tula.net (M.A. Chebotar). 1 Partially supported by a grant from the Ministry of Science of Slovenia /00/$ - see front matter 2000 Published by Elsevier Science Inc. All rights reserved. PII:S (00)

2 198 K.I. Beidar et al. / Linear Algebra and its Applications 312 (2000) The study of Jordan isomorphisms of associative rings and algebras, primarily concerned with their relations to (associative) (anti)isomorphisms, goes back to Ancochea [1,2], Kaplansky [12], Hua [10], Jacobson and Rickart [11], Herstein [9] and Smiley [19]. More recently, some further progress has been made by Baxter and Martindale [3], McCrimmon [18], Brešar [6,7], Lagutina [13], Martindale [16] and Beidar et al. [4]. In all these papers, some structural properties of certain classes of rings and algebras have been used, and apparently none of them cover the problem of describing Jordan isomorphisms of T r (C), the algebra of all r r upper triangular matrices over a commutative ring C (we remark that as simple as this algebra appears to be, from the structural point of view it is incomparably more complicated than, say, the algebra of all r r matrices). Using linear algebraic techniques, Molnár and Šemrl [17] recently proved that automorphisms and anti-automorphisms are the only Jordan automorphisms of T r (F),whereF is a field with at least three elements. On the other hand, Dokovič [8] considered an analogous problem concerning Lie automorphisms on triangular matrix algebras (see also [14,15]). The main result in [8] characterizes Lie automorphisms of T r (C), wherec is a commutative ring with 1 which is connected, i.e., a ring in which the only idempotents are 0 and 1. The goal of the present paper is to obtain a Jordan analogue of this Lie type result, which also generalizes the theorem of Molnár and Šemrl. Our result is as follows. Theorem. Let C be a 2-torsionfree commutative ring with identity. The following two conditions are equivalent: (i) C is a connected ring; (ii) every Jordan isomorphism of C-algebra T r (C), r 2, onto an arbitrary C-algebra is either an isomorphism or an anti-isomorphism. The condition that C is 2-torsionfree means that 2A = 0 with A T r (C), implies A = 0. We remark that in the case when 2T r (C) = 0 the concept of a Jordan isomorphism coincides with that of a Lie isomorphism, so that for this case we can refer to [8]. Let us also mention another reason for our interest in Jordan isomorphisms of upper triangular matrix algebras. The need to know their form appears in our forthcoming paper [5] which is devoted to functional identities on these algebras. Before proving the theorem we first give some general comments on Jordan isomorphisms. Let T and A be 2-torsionfree algebras over a commutative ring C.Then every Jordan isomorphism ϕ : T A clearly satisfies ϕ(a 2 ) = ϕ(a) 2 for all A T. Further, from 2ABA = A(AB + BA) + (AB + BA)A (A 2 B + BA 2 ) we see that ϕ also satisfies ϕ(aba) = ϕ(a)ϕ(b)ϕ(a) for all A, B T. This obviously yields ϕ(abc + CBA) = ϕ(a)ϕ(b)ϕ(c) + ϕ(c)ϕ(b)ϕ(a) for all A, B, C T. Now let E T be an idempotent. Suppose that AE = EA = 0forsomeA T. In particular, EA + AE = 0 = EAE and so ϕ(e)ϕ(a) + ϕ(a)ϕ(e) = 0 = ϕ(e)ϕ(a)ϕ(e). However,ϕ(E) is an idempotent in A and so these two identities

3 K.I. Beidar et al. / Linear Algebra and its Applications 312 (2000) readily imply ϕ(a)ϕ(e) = ϕ(e)ϕ(a) = 0. Since ϕ 1 is also a Jordan isomorphism, the same argument shows that the converse is also true. That is, given A, E T with E 2 = E, wehave EA = AE = 0 ϕ(e)ϕ(a) = ϕ(a)ϕ(e) = 0. (1) Proof of the theorem. Assume first that C is not connected, i.e., there is an idempotent ε C different from 0 and 1. From the observation at the beginning of this note it is clear that in order to construct a Jordan automorphism of T r = T r (C) which is neither an automorphism nor an anti-automorphism it is enough to find an anti-automorphism of T r. They do exist indeed. For example, the map A UA tr U 1 is an anti-automorphism (as a matter of fact, even an involution) of T r,wherea tr denotes the transpose of A and U = E 1r + E 2r 1 + +E r 12 + E r1, cf. [8,14,17] (here, E ij denotes a matrix unit). Assume now that C is connected and that ϕ is a Jordan isomorphism of T r onto an algebra A over C. We must show that ϕ is either an isomorphism or an antiisomorphism. According to (1) we have E ii A = AE ii = 0 ϕ(e ii )ϕ(a) = ϕ(a)ϕ(e ii ) = 0 (2) holds true for all A T r and i = 1,...,r. This is the key observation, as we shall see. We proceed to prove the theorem by induction on r. Soletfirstr = 2. Set e = ϕ(e 11 ), f = ϕ(e 22 ) and n = ϕ(e 12 ). Clearly, e 2 = e, f 2 = f and n 2 = 0. Further, applying (2) we see that ef = fe = 0, and from the identities E 12 = E 11 E 12 + E 12 E 11 = E 22 E 12 + E 12 E 22 and E 11 E 12 E 11 = E 22 E 12 E 22 = 0weinfern = en + ne = fn+ nf and ene = fnf = 0. Using these relations it is easy to see that ϕ is an isomorphism if and only if n = en (i.e., when ne = 0), and ϕ is an anti-isomorphism if and only if n = ne (i.e., when en = 0). Thus, it suffices to show that either en = 0oren = n.sinceϕ is a linear isomorphism, the elements e, f, n A span the C-module A. Therefore, en = λe + µf + νn for some λ, µ, ν C.However,multiplying this relation from the left- and right-hand sides by e and using ene = ef e = 0wegetλe = 0. Similarly, we see that µf = 0. Hence, en = νn which yields νn = en = e(en) = e(νn) = ν 2 n.thatis,ϕ((ν 2 ν)e 12 ) = 0andsoν 2 = ν. However, C is assumed to be connected, so it follows that ν = 0orν = 1. That is, either en = 0 or en = n. This proves the theorem for r = 2. Now let r 3. Let U be the set of all matrices A = (a ij ) 1 i j r T r such that a 11 = a 12 = =a 1r = 0. That is, U consists of matrices whose first row is zero. Similarly, let V be the set of all matrices in T r whose rth column is zero (i.e., a 1r = a 2r = =a rr = 0). Note that U and V are subalgebras of T r (moreover, U is a right and V is a left ideal of T r ), and they both are isomorphic to T r 1 (C). Clearly, A T r belongs to U if and only if AE 11 = E 11 A = 0 (actually, the condition E 11 A = 0 is sufficient, but we need both relations in order to apply (2)). Similarly, A T r belongs to V if and only if AE rr = E rr A = 0. Now, in view of (2), an element a A lies in ϕ(u) if and only if aϕ(e 11 ) = ϕ(e 11 )a = 0, and a A lies in ϕ(v) if and only if aϕ(e rr ) = ϕ(e rr )a = 0. From this we clearly

4 200 K.I. Beidar et al. / Linear Algebra and its Applications 312 (2000) infer that ϕ(u) and ϕ(v) are (associative) subalgebras of A. Therefore, the restriction of ϕ to U is a Jordan isomorphism between associative algebras U and ϕ(u). Since U is isomorphic to T r 1 (C), the induction assumption can be used. Therefore, the restriction of ϕ to U is either an isomorphism or an anti-isomorphism. Similarly, the same is true for the restriction of ϕ to V. Suppose, for instance, that ϕ is an isomorphism on U and an anti-isomorphism on V. Since E 13 = E 12 E 22 E 23 + E 23 E 22 E 12, we have ϕ(e 13 ) = ϕ(e 12 )ϕ(e 22 )ϕ(e 23 ) + ϕ(e 23 )ϕ(e 22 )ϕ(e 12 ). However, since E 23 = E 23 E 33 and ϕ is an isomorphism on U, we have ϕ(e 23 ) = ϕ(e 23 )ϕ(e 33 ) and hence ϕ(e 23 )ϕ(e 22 )ϕ(e 12 ) = 0 for ϕ(e 33 )ϕ(e 22 ) = 0 by (2). Similarly, since ϕ is an anti-isomorphism on V it follows that ϕ(e 12 ) = ϕ(e 12 )ϕ(e 11 ) which yields ϕ(e 12 )ϕ(e 22 )ϕ(e 23 ) = 0. But then ϕ(e 13 ) = 0, a contradiction. Similarly, we see that ϕ cannot be an anti-isomorphism on U and an isomorphism on V. Assume now that ϕ is an isomorphism on both U and V, and let us show that in this case ϕ is an isomorphism on the whole T r. Let us first show that ϕ(e ij E 1r ) = ϕ(e ij )ϕ(e 1r ) (3) for all 1 i j r.sinceϕ is a Jordan isomorphism, (3) is equivalent to ϕ(e 1r E ij ) = ϕ(e 1r )ϕ(e ij ). (4) We have to consider different possibilities. First, if 2 j r 1, then ϕ(e ij ) = ϕ(e ij E jj ) = ϕ(e ij )ϕ(e jj ) for both E ij,e jj lie in V. But then (2) implies ϕ(e ij )ϕ(e 1r ) = 0 which proves (3) for this case. Similarly, we see that (4) is fulfilled when 2 i r 1. Thus, it remains to consider the following three possibilities: i = j = 1, i = j = r, and i = 1, j = r. The latter one is trivial since E1r 2 = 0. So, let i = j = 1. First, note that ϕ(e 1r) = ϕ(e 12 E 2r + E 2r E 12 ) = ϕ(e 12 )ϕ(e 2r ) + ϕ(e 2r )ϕ(e 12 ).Sinceϕ is an isomorphism on U this further implies that ϕ(e 1r ) = ϕ(e 12 )ϕ(e 2r )ϕ(e rr ) + ϕ(e 2r )ϕ(e 12 ). But this, together with ϕ(e rr )ϕ(e 11 ) = 0 (recall (2)) and ϕ(e 12 )ϕ(e 11 ) = ϕ(e 12 E 11 ) = 0 (namely, ϕ is an isomorphism on V) implies ϕ(e 1r )ϕ(e 11 ) = 0, proving (4). Similarly, we consider the last remaining case when i = j = r. Thus, (3) (and thereby (4)) is proved. All we still need to show is that ϕ(e 1i E jr ) = ϕ(e 1i )ϕ(e jr ) (5) for all 1 i r 1, 2 j r. Clearly, (5) is equivalent to ϕ(e jr E 1i ) = ϕ(e jr )ϕ(e 1i ). (6) Since E jr,e rr U and E 11,E 1i V, we have ϕ(e jr ) = ϕ(e jr )ϕ(e rr ) and ϕ(e 1i ) = ϕ(e 11 )ϕ(e 1i ).Butthenϕ(E jr )ϕ(e 1i ) = 0 by (2). This proves (6) (and thereby (5)). Thus, ϕ is indeed an isomorphism on T r. One completes the proof by showing in a similar manner that ϕ must be an antiisomorphism in the case when its restrictions to both U and V are anti-isomorphisms.

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