Additive Maps on Units of Rings

Košan, Tamer ; Sahinkaya, Serap ; Zhou, Yiqiang

Canadian mathematical bulletin, Tome 61 (2018) no. 1, pp. 130-141

Résumé

Let $R$ be a ring. A map $f\,:\,R\,\to \,R$ is additive if $f(a\,+\,b)\,=\,f(a)\,+\,f(b)$ for all elements $a$ and $b$ of $R$ . Here, a map $f\,:\,R\,\to \,R$ 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 $\text{Xu}$ , $\text{Pei}$ and $\text{Yi}$ 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 of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to ${{\mathbb{Z}}_{2}}\,\text{or}\,R/J(R)\,\cong \,{{\mathbb{Z}}_{2}}\,$ 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 unithomomorphic 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.

Détail

DOI : 10.4153/CMB-2017-019-3

Mots-clés : 15A99, 16U60, 16L30, additivemap, unit, 2-sum property, semilocal ring, exchange ring with primitive factors Artinian