matrix ring

0.1 Matrix Rings

A ring R is said to be a matrix ring if there is a ring S and a positive integer n such that


the ring of n×n matrices with entries as elements of S. Usually, we simply identify R with Mn(S).

Generally, one is interested to find out if a given ring R is a matrix ring. By setting n=1, we see that every ring is trivially a matrix ring. Therefore, to exclude these trivial cases, we call a ring R a trivial matrix ring if there does not exist an n>1 such that RMn(S). Now the question becomes: is R a non-trivial matrix ring?

Actually, the requirement that S be a ring in the above definition is redundent. It is enough to define S to be simply a set with two binary operationsMathworldPlanetmath + and . Fix a positive integer n1, define the set of formal n×n matrices Mn(S) with coefficients in S. Addition and multiplication on Mn(S) are defined as the usual matrix additionMathworldPlanetmath and multiplication, induced by + and of S respectively. By abuse of notation, we use + and to denote addition and multiplication on Mn(S). We have the following:

  1. 1.

    If Mn(S) with + is an abelian groupMathworldPlanetmath, then so is S.

  2. 2.

    If in addition, Mn(S) with both + and is a ring, then so is S.

  3. 3.

    If Mn(S) is unital (has a multiplicative identityPlanetmathPlanetmath), then so is S.

The first two assertions above are easily observed. To see how the last one roughly works, assume E is the multiplicative identity of Mn(S). Next define U(a,i,j) to be the matrix whose (i,j)-cell is aS and 0 everywhere else. Using cell entries est from E, we solve the system of equations


to conclude that E takes the form of a diagonal matrixMathworldPlanetmath whose diagonal entries are all the same element eS. Furthermore, this e is an idempotentMathworldPlanetmathPlanetmath. From this, it is easy to derive that e is in fact a multiplicative identity of S (multiply an element of the form U(a,1,1), where a is an arbitrary element in S). The converse of all three assertions are clearly true too.


  • It can be shown that if R is a unital ring having a finite doubly-indexed set T={eij1i,jn} such that

    1. (a)

      eijek=δjkei where δjk denotes the Kronecker deltaMathworldPlanetmath, and

    2. (b)


    then R is a matrix ring. In fact, RMn(S), where S is the centralizerMathworldPlanetmath of T.

  • A unital matrix ring R=Mn(S) is isomorphicPlanetmathPlanetmathPlanetmath to the ring of endomorphisms of the free modulePlanetmathPlanetmath Sn. If S has IBN, then Mn(S)Mm(S) implies that n=m. It can also be shown that S has IBN iff R does.

  • Any ring S is Morita equivalent to the matrix ring Mn(S) for any positive integer n.

0.2 Matrix Groups

Suppose R=Mn(S) is unital. U(R), the group of units of R, being isomorphic to the group of automorphismsPlanetmathPlanetmathPlanetmath of Sn, is called the general linear groupMathworldPlanetmath of Sn. A matrix group is a subgroupMathworldPlanetmathPlanetmath of U(R) for some matrix ring R. If S is a field, in particular, the field of real numbers or complex numbers, matrix groups are sometimes also called classical groups, as they were studied as far back as the 1800’s under the name groups of tranformations, before the formal concept of a group was introduced.

Title matrix ring
Canonical name MatrixRing
Date of creation 2013-03-22 15:54:18
Last modified on 2013-03-22 15:54:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 16S50
Defines matrix group