Morita equivalence for graded rings

TL;DR

This paper extends Morita equivalence to graded rings, establishing the equivalence of categorical, isomorphic, and bimodule conditions in the graded setting, including infinite matrix ring isomorphisms.

Contribution

It introduces a graded version of Morita equivalence, connecting various graded algebraic structures and their categorical and isomorphic properties.

Findings

Proves graded Morita equivalence conditions are equivalent.

Extends classical Morita theorems to graded rings.

Includes infinite matrix ring isomorphisms in the graded context.

Abstract

The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.