In the context of a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 M over a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 R, the top of M is the largest semisimple

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...

 quotient module

Quotient module

In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic...

 of M if it exists.

For finite-dimensional k-algebras (k a field), if rad(M) denotes the intersection of all proper maximal submodules of M (the radical of the module

Radical of a module

In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc of M.-Definition:Let R be a ring and M a left...

), then the top of M is M/rad(M). In the case of local rings with maximal ideal P, the top of M is M/PM. In general if R is a semilocal ring (=semi-artinian ring), that is, if R/Rad(R) is an Artinian ring

Artinian ring

In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...

, where Rad(R) is the Jacobson radical

Jacobson radical

In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...

of R, then M/rad(M) is a semisimple module and is the top of M. This includes the cases of local rings and finite dimensional algebras over fields.