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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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