Introduction. In §2 of Chapter I, we introduced, as an illustration of the concept of ring, the ring of all real matrices of order 2, and suggested that this example could be generalized in at least two different ways. In the present chapter we shall make these generalizations and study matrices in some detail. The necessary definitions will be given presently, but first we make a few remarks which will help to put the rest of the chapter in proper perspective.

The importance of the theory of matrices stems from at least two facts. In the first place, the theory furnishes a convenient approach to the study of linear transformations, which are fundamental in many branches of mathematics. In the second place, it turns out that a great many rings are isomorphic to suitably chosen rings of matrices with, say, real or complex elements, and therefore a study of such rings is of quite general interest. Most of the extensive theory of matrices deals with matrices whose elements are from a given field or integral domain. These important cases are well covered by the readily available texts listed at the end of this chapter, in which the reader will also find adequate treatment of the important aspects of the theory just mentioned. Accordingly, we confine our remarks to a more general situation, and in the following section shall introduce matrices with elements in an arbitrary ring and mention a few fundamental properties. Beginning with §37, the matrices considered will have elements in an arbitrary commutative ring with unit element.