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