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.

Keywords

• Krull-Schmidt Theorem
• indecomposable module
• direct-sum decomposition
• endomorphism ring
• uniserial module

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