Matrix ring

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full matrix ring

The ring of all square matrices of a fixed order over a ring . The ring of -dimensional matrices over is denoted by or . Throughout this article is an associative ring with identity (cf. Associative rings and algebras).

The ring is isomorphic to the ring of all endomorphisms of the free right -module , possessing a basis with elements. The matrix is the identity in . An associative ring with identity 1 is isomorphic to if and only if there is in a set of elements , , subject to the following conditions:

1) , ;

2) the centralizer of the set of elements in is isomorphic to .

The centre of coincides with , where is the centre of ; for the ring is non-commutative.

The multiplicative group of the ring (the group of all invertible elements), called the general linear group, is denoted by . A matrix from is invertible in if and only if its columns form a basis of the free right module of all -dimensional matrices over . If is commutative, then the invertibility of a matrix in is equivalent to the invertibility of its determinant, , in . The equality holds.

The ring is simple if and only if is simple, for the two-sided ideals in are of the form , where is a two-sided ideal in . An Artinian ring is simple if and only if it is isomorphic to a matrix ring over a skew-field (the Wedderburn–Artin theorem). If denotes the Jacobson radical of the ring , then . Consequently, every matrix ring over a semi-simple ring is semi-simple. If is regular (i.e. if for every there is a such that ), then so is . If is a ring with an invariant basis number, i.e. the number of elements in a basis of each free -module does not depend of the choice of the basis, then also has this property. The rings and are equivalent in the sense of Morita (see Morita equivalence): The category of -modules is equivalent to the category of -modules. However, the fact that projective -modules are free does not necessarily entail that projective -modules are free too. For instance, if is a field and , then there exist finitely-generated projective -modules which are not free.