Semisimple module

A module is semisimple, or completely reducible, when it is a direct sum of simple modules. Equivalently, every submodule has a direct-sum complement.

For a finite-dimensional module, it suffices that every maximal proper submodule have a submodule complement in order that every submodule have one. Let $W$ be the sum of all simple modules contained in the given module. If $W$ is proper, choose a maximal submodule $M$ containing it. A complement $C$ is isomorphic to the simple quotient $V/M$ and hence is itself simple. This forces $C\subseteq W\subseteq M$, contradicting the complement. Thus $W=V$. Greedily adding simple summands not already in the sum gives a direct sum in finitely many steps. Finally, given any $U$, choose a maximal sum of these simple summands disjoint from $U$. A missing simple summand has zero intersection with their sum plus $U$, contradicting maximality unless their sum complements $U$. This proves the semisimple module criterion, including the zero module.

Suppose the $S_i$ are pairwise nonisomorphic simple modules. Any submodule $W$ of their finite direct sum that projects onto every summand is the entire sum. Induct on $k$. Projection onto the first $k-1$ factors is onto by induction. Its kernel is a submodule of $S_k$, hence zero or all of $S_k$. If zero, $W$ is isomorphic to the first $k-1$ factors but also has quotient $S_k$, contradicting their distinct composition factors and the Jordan–Hölder theorem. Therefore the kernel is $S_k$, proving the assertion. Consequently, if $S_i=A{x}_i$, the map $a\mapsto (a{x}_i{)}_i$ is onto and gives $A/({\bigcap }_i{\mathrm{Ann}}_A(x_i))\cong {⨁}_i{S}_i$ as left modules. The annihilators of these vectors need only be left ideals.

A finite sum of simple modules inside one module is a semisimple module. Inductively, if $M$ is the sum already retained, then $M\cap S_j$ is a submodule of the simple module $S_j$, hence zero or $S_j$. In the first case append $S_j$ as a direct summand; in the second it adds nothing. This gives a direct sum of modules selected from the original family.

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.

In a complex representation of a finite group $N$, the $\theta$-isotypic component is the sum of all irreducible submodules with character $\theta$. The character idempotent $e_{\theta }=\theta (1)|N{|}^{-1}{\sum }_{n\in N}\theta (n^{-1})n$ projects onto it. Every $N$-submodule is preserved by these projections and decomposes into its intersections with the isotypic components. If $N◃G$, conjugation permutes them according to ${\theta }^g(n)=\theta (g^{-1}ng)$.