Tensor-Hom formalism for modules over nonunital rings

TL;DR

This paper introduces and explores the concepts of t-unital and c-unital rings and modules, establishing their categorical properties, relations, and implications for nonunital ring theory and semialgebras over coalgebras.

Contribution

It defines t-unital and c-unital modules, proves their categories are abelian and equivalent, and analyzes their properties and behavior under ring homomorphisms.

Findings

Categories of t-unital and c-unital modules are abelian and equivalent.

Full subcategories of s-unital modules form hereditary torsion classes.

The paper connects these concepts to semialgebras over coalgebras.

Abstract

We say that a ring $R$ is t-unital if the natural map $R\otimes_RR\rightarrow R$ is an isomorphism, and a left $R$-module $P$ is c-unital if the natural map $P\rightarrow\operatorname{Hom}_R(R,P)$ is an isomorphism. For a t-unital ring $R$, the category of t-unital left $R$-modules is a unital left module category over an associative, unital monoidal category of t-unital $R$-$R$-bimodules, while the category of c-unital left $R$-modules is opposite to a unital right module category over the same monoidal category. Any left s-unital ring $R$, as defined by Tominaga in 1975, is t-unital; and a left $R$-module is s-unital if and only if it is t-unital. For any (nonunital) ring $R$, the full subcategory of s-unital $R$-modules is a hereditary torsion class in the category of nonunital $R$-modules; and for rings $R$ arising from small preadditive categories, the full subcategory of c-unital $R$-modules is closed under kernels and cokernels. However, over a t-unital ring $R$, the full subcategory of t-unital modules need not be closed under kernels, and the full subcategory of c-unital modules need not be closed under cokernels in the category of nonunital modules. Nevertheless, the categories of t-unital and c-unital left $R$-modules are abelian and naturally equivalent to each other; they are also equivalent to the quotient category of the abelian category of nonunital modules by the Serre subcategory of modules with zero action of $R$. This is a particular case of the result from a 1996 manuscript of Quillen. We also discuss related flatness, projectivity, and injectivity properties; and study the behavior of t-unitality and c-unitality under the restriction of scalars for a homomorphism of nonunital rings. Our motivation comes from the theory of semialgebras over coalgebras over fields.