Definition 4

Let A be a ring, M an A-module, and S a simple A-module. The isotypical component of type S of M, denoted by $M_S$, is the sum of the submodules of M isomorphic to S.

It is clear that $M_S$ is the greatest submodule of M that is isotypical of type S. Because every submodule of $M_S$ is isotypical of type S (VIII, p. 61, Proposition 2), we have $N_S= M_S\cap N$ for every submodule N of M.

If $S'$ is a simple A-module isomorphic to S, then we clearly have $M_S=$ $M_{S'}$, so $M_S$ depends only on the class of S (VIII, p. 51).

Let M be an A-module. There exists a greatest semisimple submodule of M, called the socle of M; it is the sum of the simple submodules of M and also the sum of the isotypical components of M. In particular, M is semisimple if and only if it is equal to its socle.