Tensor-Hom formalism for modules over nonunital rings

Abstract

We say that a ring $R$ is t-unital if the natural map $R{\otimes }_{R}R\to R$ is an isomorphism, and a left $R$-module $P$ is c-unital if the natural map $P\to {\mathrm{Hom}}_R(R,P)$ 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.

arXiv commentLaTeX 2e with xy-pic; 41 pages, 8 commutative diagrams; v.2: a reference added to the effect that Theorem 2.6 is due to Tominaga; Remark 1.8, Example 2.11, and Section 5 added; v.3: abstract rewritten, order of sections rearranged, new Example 4.2 inserted, new Sections 5, 7, and 8 added, many references to Quillen's manuscript added; v.4: explanations added in Section 5; new Section 9 added