The similarity submodules for various lattices and modules of interest in the crystal and quasicrystal theory are determined and the corresponding semigroups, along with their zeta functions, are investigated. One application is to the colouring of crystals and quasicrystals.

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...

An (A,B)-cyclic submodule M is generated by the states of one single trajectory of a linear control system whose parameters come from a commutative ring. M is “finite”, when it is generated by the states of a “deadbeat-control” process. Motivations and basic properties of such modules are given and among several further results it is shown that the family of finite (A,B)-cyclic submodules is an...

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

In this paper we study totally projective QTAG-modules and the extensions of bounded QTAG-modules. In the first section we study totally projective modules M/N and M ′/N ′ where N , N ′ are isomorphic nice submodules of M and M ′ respectively. In fact the height preserving isomorphism between nice submodules is extented to the isomorphism from M onto M ′ with the help of Ulm-Kaplansky invariant...