Uniqueness of Decomposition, Factorisations, G-Groups and Polynomials

Abstract

In this article, we present the classical Krull-Schmidt Theorem for groups, its statement for modules due to Azumaya, and much more modern variations on the theme, like the so-called weak Krull-Schmidt Theorem, which holds for some particular classes of modules. Also, direct product of modules is considered. We present some properties of the category of G-groups, a category in which Remak's results about the Krull-Schmidt Theorem for groups can be better understood. In the last section, direct-sum decompositions and factorisations in other algebraic structures are considered.

Algebraic structures from the point of view of complete multiplicative lattices

“…Another immediate application of our theory is to G-groups, which we studied in [1,4,5,15,16]. The notion of G-group H is classical.…”