Текшеуге келтірілетін модульдердің тікелей қосындысы

Direct sum of irreducible modules
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 that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group theory analog of the same notion, see Semisimple representation.

If M is semisimple and N is a submodule, then N and M / N are also semisimple. An arbitrary direct sum of semisimple modules is semisimple. A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero.

A semisimple module M over a ring R can also be thought of as a ring homomorphism from R into the ring of abelian group endomorphisms of M. The image of this homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image. The endomorphism ring of a semisimple module is not only semiprimitive, but also von Neumann regular.

For a commutative ring, the four following properties are equivalent: being a semisimple ring; being artinian and reduced; being a reduced Noetherian ring of Krull dimension 0; and being isomorphic to a finite direct product of fields. If K is a field and G is a finite group of order n, then the group ring K[G] is semisimple if and only if the characteristic of K does not divide n. This is Maschke's theorem, an important result in group representation theory. By the Wedderburn–Artin theorem, a unital ring R is semisimple if and only if it is (isomorphic to) Mn1(D1) × Mn2(D2) × × Mnr(Dr), where each Di is a division ring and each ni is a positive integer, and Mn(D) denotes the ring of n by n matrices with entries in D.
An example of a semisimple non unital ring is M∞(K), the row finite, column finite, infinite matrices over a field K.

A ring is called Jacobson semisimple (or J semisimple or semiprimitive) if the intersection of the maximal left ideals is zero, that is, if the Jacobson radical is zero. Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J semisimple ring is semisimple if and only if it is an artinian ring, so semisimple rings are often called artinian semisimple rings to avoid confusion. For example, the ring of integers, Z, is J semisimple, but not artinian semisimple.