Semisimple rings

A semisimple ring is an algebraic structure where every module (representation) can be broken down into simple, indivisible parts without losing information, akin to how complex objects can be deconstructed into basic building blocks. More formally, it is a ring in which every module is a direct sum of simple modules—those with no further non-trivial submodules. This property ensures a well-understood, highly organized structure similar to prime factorization, making semisimple rings fundamental in algebra for studying modules, representations, and the classification of algebraic systems.