Cannot be dismantled

In the mathematical sub-area of algebra , an indivisible module is a module that cannot be broken down into a direct sum . One can show that every module that fulfills certain requirements is a direct sum of indecomposable modules (see: Theorem of Krull-Remak-Schmidt ). However, there are also rings and modules for which this is not the case.

definition

A module over a ring is called indivisible if it can not be written as the direct sum of two non-zero modules and . ${\displaystyle R}$ ${\displaystyle M\neq 0}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$

This definition can be applied to any Abelian category .

Examples

• A vector space over a body is indecomposable if and only if it is one-dimensional. ${\displaystyle K}$ ${\displaystyle K}$
• Every simple module ${\displaystyle R}$ cannot be dismantled, but not the other way around.
• A module of finite length is indecomposable if and only if its endomorphism ring is local .

Individual evidence

1. ↑ Jens Averdunk: Modules with additional properties / Jens Averdunk . Utz, Wiss., Munich 1997, ISBN 3-89675-184-0 , p. 15 .