Morita Theory for Rings and Semigroups

Abstract

The notion of Morita equivalence for rings defines a relationship between rings in terms of their module categories being equivalent in the sense of category theory. To characterise Morita equivalence for rings, Morita contexts and factors on various bimodules have emerged. As a generalisation of Morita equivalence, the concept of Morita-like equivalences was developed to investigate xst-rings. The study of Morita invariants is also an important branch in the Morita theory for rings. Analogous to the Morita theory for rings, Morita equivalence and Morita invariants for semigroups have been developed. Four major approaches to the characterisations of Morita equivalence between semigroups have appeared. They are categories of acts over semigroups, Morita contexts, Cauchy completions and enlargements.The aim of this article is to make a brief survey of Morita equivalence for rings not necessary with an identity and semigroups.