Dual Kasch Rings

TL;DR

This paper introduces and characterizes the concept of right dual Kasch rings, exploring their properties, examples, and relationships with other ring classes, extending the classical Kasch ring theory.

Contribution

It defines dual Kasch rings, provides characterizations, examples, and explores their behavior under various ring constructions and conditions.

Findings

Self-injective, V-, and commutative perfect rings are dual Kasch.

Skew group rings of dual Kasch rings are dual Kasch if the group order is invertible.

Commutative Kasch rings are dual Kasch; finite Goldie dimension rings are dual Kasch iff classical.

Abstract

It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field $k$, a finite dimensional $k$-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.