Simple maximal quotient rings

Abstract:

In this paper we consider the question of when a ring $\Lambda$ has a simple maximal left ring of quotients. In the first section we determine two necessary conditions; viz. that $\Lambda$ be left nonsingular, and when $I$ and $J$ are nonzero ideals of $\Lambda$ with $I \cap J = 0$, then $I + J$ is not left essential in $\Lambda$. In the second section we show that these conditions are also sufficient when $\Lambda$ is of finite left Goldie dimension. In addition, for a left nonsingular ring of finite left Goldie dimension, we determine the ideal structure of the maximal left ring of quotients.