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 ⊆ 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 ∈ 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 ∈ I there is a non-zeroWe 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, ACA = 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.

On the Galois map for groupoid actions

“…Since the work published in 1926 by H. Brandt in [1], the study of the theory of groupoids is growing and gaining developments in different areas of mathematics and even physics, as shown in studies involving algebraic topology, noncommutative geometry, Lie groupoids, partial actions and Galois theory ( [3], [4], [5], [6], [8], [13], [14], [15]).…”

Semigroup graded algebras and graded PI-exponent

“…Earlier results on semigroup graded rings that satisfy a polynomial identity have been obtained by Kelarev [25] and Clase and Jespers [9,10]. A characterisation of semigroup graded rings that are simple has received a lot of attention, see for example [5,4,6,24,29]. For more results on graded rings we refer to [26].…”