Abstract view
Additive maps on units of rings


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
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 unitadditive 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
unitadditive map of ${\mathbb M}_n(F)$ is additive for all $n\ge
2$, this paper is about the question when every unitadditive
map of a ring is additive. It is proved that every unitadditive
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 unitadditive 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 unitadditive 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
unithomomorphic map of a ring is an endomorphism is addressed.