Simple rings and degree maps

Abstract:
For an extension $A/B$ of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of $A$ with a property that we call $A$-simplicity of $B$. By this we mean that there is no non-trivial ideal $I$ of $B$ being $A$-invariant, that is satisfying $AI \subseteq IA$. We show that $A$-simplicity of $B$ is a necessary condition for simplicity of $A$ for a large class of ring extensions when $B$ is a direct summand of $A$. To obtain sufficient conditions for simplicity of $A$, we introduce the concept of a degree map for $A/B$. By this we mean a map $d$ from $A$ to the set of non-negative integers satisfying the following two conditions: (d1) if $a \in A$, then $d(a) = 0$ if and only if $a=0$; (d2) there is a subset $X$ of $B$ generating $B$ as a ring such that for each non-zero ideal $I$ of $A$ and each non-zero $a \in I$ there is a non-zero $a' \in I$ with $d(a') \leq d(a)$ and $d(a'b - ba') < d(a)$ for all $b \in X$. We show that if the centralizer $C$ of $B$ in $A$ is an $A$-simple ring, every intersection of $C$ with an ideal of $A$ is $A$-invariant, $A C A = A$ and there is a degree map for $A/B$, then $A$ is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings.