Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Separable functors and firm modules

Patrik Lundström

Abstract

We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.

Separable functors and firm modules

Abstract

We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.
Paper Structure (4 sections, 38 theorems, 29 equations)

This paper contains 4 sections, 38 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Separable functors
  3. Module categories
  4. Semisimple modules

Key Result

Theorem 1

Suppose that $f : B \to A$ is a ring homomorphism of unital rings. Then $A/B$ is separable if and only if the restriction functor ${\rm Res} : {}_A {\rm Mod} \to {}_B {\rm Mod}$ is separable.

Theorems & Definitions (75)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 65 more