Semiendomorphisms of simple near-rings

Abstract

Let $N$ be a finite simple centralizer near-ring that is not an exceptional near-field. A semiendomorphism of $N$ is a map ’ from $N$ into $N$ such that $(a + b)’ = a’ + b’,(aba)’ = a’b’a’$, and $1’ = 1$ for all $a,b \in N$. It is shown that every semiendomorphism of $N$ is an automorphism of $N$. A Jordan-endomorphism of $N$ is a map ’ from $N$ into $N$ such that $(a + b)’ = a’ + b’,(ab + ba)’ = a’b’ + b’a’$, and $1’ = 1$ for all $a,b \in N$. It is shown that every Jordan-endomorphism of $N$ is an automorphism assuming $2 \in N$ is invertible. The above results imply that every semiendomorphism (Jordan-endomorphism) of a "special" class of semisimple near-rings is an automorphism. These results are in contrast to the ring situation where semiendomorphisms tend to be either an automorphism or an antiautomorphism.