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, \"{O}inert and Royer, and then, in turn, for Steinberg algebras, over non-commutative rings, by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michel.