Simplicity of algebras via epsilon-strong systems

Abstract

We obtain sufficient criteria for simplicity of systems, that is, rings $R$ that are equipped with a family of additive subgroups $R_s$, for $s \in S$, where $S$ is a semigroup, satisfying $R = \sum_{s \in S} R_s$ and $R_s R_t \subseteq R_{st}$, for $s,t \in S$. These criteria are specialized to obtain sufficient criteria for simplicity of, what we call, s-unital epsilon-strong systems, that is systems where $S$ is an inverse semigroup, $R$ is coherent, in the sense that for all $s,t \in S$ with $s \leq t$, the inclusion $R_s \subseteq R_t$ holds, and for each $s \in S$, the $R_s R_{s^*}$-$R_{s^*}R_s$-bimodule $R_s$ is s-unital. As an aplication of this, we obtain generalizations of recent criteria for simplicity of skew inverse semigroup rings, by Beuter, Goncalves, Öinert and Royer, and then, in turn, for Steinberg algebras, over non-commutative rings, by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michel.