Inverse rings and inverse semigroups of ring homomorphisms

Abstract

In this paper, a ring is an inverse ring if its multiplicative semigroup is an inverse semigroup. We show that an inverse ring contains no nilpotent elements except 0 and that it is a subring of a subdirect product of skew fields.

Let $R=Z_{n}$. Let $(H(R),\circ )$ be the semigroup of ring homomorphisms(under composition) on $R$. We show that $H(R)$ is a commutative inverse semigroup and it is of order $2^{n}$ and that each of its elements has order 2 or less.

We show that the set of regular-ring homomorphisms on $%Z_{p}[x]$, where $p$ is a prime, is an inverse semigroup.