Matrix ring

In abstract algebra the matrix ring M(n, R) is the set of all n×n matrices over an arbitrary ring R. This set is itself a ring under matrix addition and multiplication.

[edit] Properties

The matrix ring M(n, R) is commutative if and only if n ≤ 1 or R is the trivial ring.
The center of a matrix ring over R consists of the matrices which are of the form b times the identity matrix, where b belongs to the center of R.
A matrix ring over a division ring is an Artinian simple ring. The converse is also true and called the Artin-Wedderburn theorem.
A matrix ring over a prime ring is a prime ring.
The matrix ring M(n, R) can be identified with the endomorphism ring End(Rⁿ).
There is a one-to-one correspondence between the (two-sided) ideals of M(n, R) and the (two-sided) ideals of R. Namely, for each ideal I of R, the set of all n-by-n matrices with entries in I is an ideal of M(n, R), and each ideal of M(n, R) arises in this way. This implies that M(n, R) is simple if and only if R is simple. However, this correspondence is not one-to-one if we consider right ideals (or left ideals) rather than two-sided ideals. For example, M(n, R) has the proper nonzero right ideal consisting of all matrices whose columns are zero except for possibly the first column, but the set of all entries of matrices from this ideal is equal to R. The correspondence for left and right ideals is rather with submodules of free modules.