 Isotypic decomposition

A finite-length semisimple module has a canonical decompositionin which $M_i$ is the sum of all simple submodules isomorphic to a fixed simple module $S_i$. Each $M_i$ is isomorphic to a finite direct sum $S_i^{m_i}$.

The decomposition is intrinsic to the semisimple module: the isomorphism types of its simple submodules determine the summands.