Simplicity of algebras via epsilon-strong systems
Volume 162 / 2020
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 $R_s \subseteq R_t$ for all $s,t \in S$ with $s \leq t$ and for each $s \in S$, the $R_s R_{s^*}$-$R_{s^*}R_s$-bimodule $R_s$ is s-unital. As an application, we obtain generalizations of recent criteria for simplicity of skew inverse semigroup rings by Beuter, Gonçalves, Öinert and Royer, and then for Steinberg algebras over non-commutative rings by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michell.
