matrix ring

Take any ring you trust — the integers, a field, even a noncommutative ring — and arrange its elements into n-by-n grids. Add grids entrywise and multiply them by the familiar row-times-column rule. The result is a new ring, and as soon as n is at least 2 it is noncommutative, because swapping two matrices generally changes the product. Matrix rings are the first and most important examples of noncommutativity that everyone already half-knows.

Formally, for a ring R and an integer n ≥ 1, the matrix ring M_n(R) consists of all n-by-n matrices with entries in R, under matrix addition and multiplication. It has identity the matrix with 1 on the diagonal and 0 elsewhere. Its center consists of the scalar matrices c*I with c in the center of R, so M_n of a field has center isomorphic to that field. The units of M_n(R) form the general linear group GL(n, R).

Matrix rings are the canonical simple and semisimple objects. If D is a division ring then M_n(D) is a simple ring — it has no two-sided ideals except 0 and the whole ring — and the Artin-Wedderburn theorem says these matrix rings over division rings are precisely the building blocks of every semisimple ring. The matrix units e_{ij}, having a single 1 in position (i, j), give a concrete basis and encode all the multiplication.

A subtlety to respect: the ideals of M_n(R) come entirely from ideals of R. Specifically M_n(I) is a two-sided ideal for each two-sided ideal I of R, and every two-sided ideal of M_n(R) arises this way. So M_n(R) is simple exactly when R is simple, which is why division rings (which are simple) produce simple matrix rings.