Matrix Ring - Structure

Structure

• The matrix ring M_n(R) can be identified with the ring of endomorphisms of the free R-module of rank n, M_n(R) ≅ End_R(Rn). The procedure for matrix multiplication can be traced back to compositions of endomorphisms in this endomorphism ring.

• The ring M_n(D) over a division ring D is an Artinian simple ring, a special type of semisimple ring. The rings and are not simple and not Artinian if the set is infinite, however they are still full linear rings.

• In general, every semisimple ring is isomorphic to a finite direct product of full matrix rings over division rings, which may have differing division rings and differing sizes. This classification is given by the Artin–Wedderburn theorem.

• 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×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. For n ≥ 2, not every left ideal or right ideal of M_n(R) arises by the previous construction from a left ideal or a right ideal in R. For example, the set of matrices whose columns with indices 2 through n are all zero forms a left ideal in M_n(R).

• The previous ideal correspondence actually arises from the fact that the rings R and M_n(R) are Morita equivalent. Roughly speaking, this means that the category of left R modules and the category of left M_n(R) modules are very similar. Because of this, there is a natural bijective correspondence between the isomorphism classes of the left R-modules and the left M_n(R)-modules, and between the isomorphism classes of the left ideals of R and M_n(R). Identical statements hold for right modules and right ideals. Through Morita equivalence, M_n(R) can inherit any properties of R which are Morita invariant, such as being simple, Artinian, Noetherian, prime and numerous other properties as given in the Morita equivalence article.