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Jordan isomorphisms of radical finitary matrix rings. (English) Zbl 1215.16036
A bijective map between rings is called a Jordan isomorphism, if it preserves associated Jordan multiplication \(a\circ b=ab+ba\). By I. N. Herstein’s classical theorem [Trans. Am. Math. Soc. 81, 331-341 (1956; Zbl 0073.02202)], it follows that every Jordan isomorphism between prime rings of characteristic not \(2\) is an isomorphism or anti-isomorphism; in these cases it is said to be trivial. The usual goal is to describe nontrivial Jordan isomorphisms.
In this paper under review, the authors study Jordan isomorphisms of the ring \(NT(\Gamma,K)\) of all finitary \(\Gamma\)-matrices \(\|a_{ij}\|_{i,j\in\Gamma}\), \(a_{uv}=0\), \(u\leq v\), over \(K\) for any chain \(\Gamma\) of indices. Let \(\mathcal Z_m(R)\) be \(m\)-th hyper-center of a ring \(R\). The following analogue of Herstein’s theorem is obtained. Let \(K\) be an associative ring with identity, \(\Gamma\) be a chain, \(R=NT(\Gamma,K)\) and \(J(R)=(R,+,\circ)\). Let \(R'=NT(\Omega,S)\) be chosen similarly, \(J(R)\cong J(R')\) and \(K\) has no zero-divisors. Then every Jordan isomorphism of \(R\) onto \(R'\) is trivial modulo \(\mathcal Z_3(R)\). A nontrivial Jordan isomorphism of \(R\) exists if and only if \(\mathcal Z_1(R)\neq\mathcal Z_2(R)\) and \(K\) is an integral domain of characteristic \(2\).
Reviewer: Wei Feng (Beijing)

MSC:
16W25 Derivations, actions of Lie algebras
16S50 Endomorphism rings; matrix rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16W20 Automorphisms and endomorphisms
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