Jordan Triple Elementary Maps on Rings
نویسنده
چکیده
We prove that Jordan triple elementary surjective maps on unital rings containing a nontrivial idempotent are automatically additive. The first result about the additivity of maps on rings was given by Martindale III in an excellent paper [7]. He established a condition on a ring R such that every multiplicative bijective map on R is additive. More precisely, he proved the following theorem. Theorem 1. ([7]) Let R be a ring containing a family {eα : α ∈ Λ} of idempotents which satisfies: (i) xR = {0} implies x = 0; (ii) If eαRx = {0} for each α ∈ Λ, then x = 0; (iii) For each α ∈ Λ, eαxeαR(1− eα) = {0} implies eαxeα = 0. Then any multiplicative bijective map from R onto an arbitrary ring R is additive. As a corollary, every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive. During the last decade, many mathematicians devoted to study the additivity of maps on rings as well as operator algebras. In this paper we continue to investigate the additivity of Jordan triple elementary maps on rings. We first define Jordan triple elementary maps as follows. Definition 2. Let R and R be two rings, and let M : R → R and M : R → R be two maps. Call the ordered pair (M,M) a Jordan triple elementary map of R×R if
منابع مشابه
Additivity of Jordan Elementary Maps on Rings
We prove that Jordan elementary surjective maps on rings are automatically additive. Elementary operators were originally introduced by Brešar and Šerml ([1]). In the last decade, elementary maps on operator algebras as well as on rings attracted more and more attentions. It is very interesting that elementary maps and Jordan elementary maps on some algebras and rings are automatically additive...
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