Simple Semigroup Graded Rings

We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of D. Gon\c{c}alves'. We also point out how E. Jespers' result immediately implies a generalization of a simplicity result, recently obtained by A. Baraviera, W. Cortes and M. Soares, for crossed products by twisted partial actions.Comment: 9 page

Simple Rings and Degree Maps

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, 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.Comment: 17 page

Outer partial actions and partial skew group rings

We extend the classicial notion of an outer action $\alpha$ of a group $G$ on a unital ring $A$ to the case when $\alpha$ is a partial action on ideals, all of which have local units. We show that if $\alpha$ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A \star_\alpha G$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.Comment: 16 page