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# Additive maps on units of rings

Published:2017-05-16
Printed: Mar 2018
• Tamer Koşan,
Department of Mathematics, Gebze Technical University, Gebze/Kocaeli, Turkey
• Serap Sahinkaya,
Department of Mathematics, Gebze Technical University, Gebze/Kocaeli, Turkey
• Yiqiang Zhou,
Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John's, NL A1C 5S7, Canada
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## Abstract

Let $R$ be a ring. A map $f: R\rightarrow R$ is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$ of $R$. Here a map $f: R\rightarrow R$ is called unit-additive if $f(u+v)=f(u)+f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of Xu, Pei and Yi showing that, for any field $F$, every unit-additive map of ${\mathbb M}_n(F)$ is additive for all $n\ge 2$, this paper is about the question when every unit-additive map of a ring is additive. It is proved that every unit-additive map of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong \mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${\mathbb M}_n(R)$ is additive for all $n\ge 2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unit-homomorphic if $f(uv)=f(u)f(v)$ for all units $u,v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.
 Keywords: additive map, unit, 2-sum property, semilocal ring, exchange ring with primitive factors Artinian
 MSC Classifications: 15A99 - Miscellaneous topics 16U60 - Units, groups of units 16L30 - Noncommutative local and semilocal rings, perfect rings

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