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.