In
abstract algebra, a
composition series provides a way to break up an
algebraic structure, such as a
group or a
module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not
semisimple, hence cannot be decomposed into a
direct sum of
simple modules. A composition series of a module
M is a finite increasing
filtration of
M by
submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of
M into its simple constituents.