Purity in module categories and its different generalizations have been present in the literature since the 1960s. Apart from its special role in module theory, the importance of purity increased during the last two decades through the developments of model theory and the study of locally finitely presented categories (e.g., [2, 6]). In the module theory context, that will be our framework, a p...

We describe a logarithmic tensor product theory for certain module categories for a “conformal vertex algebra.” In this theory, which is a natural, although intricate, generalization of earlier work of Huang and Lepowsky, we do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. The corresponding intertwining operators contain logarithm...

The relative homotopy theory of modules, including the (module) homotopy exact sequence, was developed by Peter Hilton (1965). Our thrust is to produce an alternative proof of the existence of the injective homotopy exact sequence with no reference to elements of sets, so that one can define the necessary homotopy concepts in arbitrary abelian categories with enough injectives and projectives, ...

The regular representation is related to Zhu’s A(V )-theory and an induced module from an A(V )-module to a V -module is defined in terms of the regular representation. As an application, a new proof of Frenkel and Zhu’s fusion rule theorem is obtained.

We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of relative test elements. They also appear as certain multiplier ideals, a concept of interest in algebraic geometry.

The regular representation is related to Zhu’s A(V )-theory and an induced module from an A(V )-module to a V -module is defined in terms of the regular representation. As an application, a new proof of Frenkel and Zhu’s fusion rule theorem is obtained.

A new kind of fuzzy module over a fuzzy ring is introduced by generalizing Yuan and Lee's definition of the fuzzy group and Aktaş and Çağman's definition of fuzzy ring. The concepts of fuzzy submodule, and fuzzy module homomorphism are studied and some of their basic properties are presented analogous of ordinary module theory.

This is a survey on the usage of the module theoretic notion of a “retractable module" in the study of algebras with actions. We explain how classical results can be interpreted using module theory and end the paper with some open questions.

This thesis presents several module systems, in particular Mei and DMei, designed for mechanized mathematics systems. Mei is a λ-calculus style module system that supports higher-order functors in a natural way. The semantics of functor application is based on substitution. A novel coercion mechanism integrates a parameter passing mechanism based on theory interpretations with simple λ-calculus...

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...