A gamma ring with minimum conditions

Abstract

The aim of this note is to study the structure of a $\Gamma$-ring (not in the sense of Nobusawa) with minimum conditions. By ring theoretical techniques, we obtain various properties on the semi-prime $\Gamma$-ring and generalize Nobusawa's result which corresponds to the Wedderburn-Artin Theorem in ring theory. Using these results, we have that a $\Gamma$-ring with minimum right and left conditions is homomorphic onto the $\Gamma_{0}$-ring $\sum_{i=1}^{q}D_{n(i),m(i)}^{(i)}$, where $D_{n(i).m(i)}^{(i)}$ is the additive abelian group of the all rectangular matrices of type $n(i)\times m(i)$ over some division ring $D^{(i)}$, and $\Gamma_{0}$ is a subdirect sum of the $\Gamma_{i}$, $1\leqq i\leqq q$, which is a non-zero subgroup of $D_{m(i),n(i)}^{(i)}$ of type $m(i)\times n(i)$ over $D^{(i)}$.