In the context of module theory and representation theory in algebra, a **semisimple module** is a specific type of module that has a particular structure. A module \( M \) over a ring \( R \) is said to be **semisimple** if it satisfies the following equivalent conditions: 1. **Direct Sum Decomposition**: \( M \) can be expressed as a direct sum of simple modules.
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A module is semisimple, or completely reducible, when it is a direct sum of simple modules. Equivalently, every submodule has a direct-sum complement.