In mathematics, the term semisimple (sometimes completely reducible) is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' ('irreducible') parts, that fit together in the cleanest way (by direct sum). Briefly, "semisimple = direct sum of simple", or equivalently "completely reducible = direct sum of irreducible".

• A semisimple module
  Semisimple module
  In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring which is a semisimple module over itself is known as an artinian semisimple ring...
  
   is one in which each submodule is a direct summand.
• A semisimple algebra
  Semisimple algebra
  In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
  
   (or ring) is one that is semisimple as a module over itself.
• A semisimple operator (or matrix) is one for which every invariant subspace has an invariant complement. This is equivalent to the minimal polynomial being square-free. Over an algebraically closed field it is equivalent to diagonalizable.
• A semisimple Lie algebra is a Lie algebra that is a direct sum of simple Lie algebras.
• A semisimple algebraic group
  Semisimple algebraic group
  In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
  
  is a linear algebraic group whose radical of the identity component is trivial.