## Tsukuba Journal of Mathematics

### A gamma ring with minimum conditions

Shoji Kyuno

#### 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 $D^{(i)}$.

#### Article information

Source
Tsukuba J. Math., Volume 5, Number 1 (1981), 47-65.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496159319

Digital Object Identifier
doi:10.21099/tkbjm/1496159319

Mathematical Reviews number (MathSciNet)
MR630521

Zentralblatt MATH identifier
0474.16030

#### Citation

Kyuno, Shoji. A gamma ring with minimum conditions. Tsukuba J. Math. 5 (1981), no. 1, 47--65. doi:10.21099/tkbjm/1496159319. https://projecteuclid.org/euclid.tkbjm/1496159319