Rings and Modules

Linear algebra deals with fields and vector spaces; here we are concerned with the generalizations to rings and modules over them. Whereas a vector space over a given field is determined up to isomorphism by its dimension, there is much greater variety for modules. Another way of regarding modules is as abelian groups, written additively, with operators. This means that much of general group theory applies, and after recalling the isomorphism theorems , proved for groups in Section 2.3, we treat a number of special situations. Semisimple modules (Section 4.3) come closest to vector spaces; they are direct sums of simple modules, but we have to bear in mind that over a given ring there may be more than one type of simple module. In the free modules (Section 4.6) we have another generalization of vector spaces. The homological treatment of module theory requires the notions of projective and injective module (Section 4.7), and they can usefully be introduced here, as they will occur again in Chapter 10, although their main use will be in FA. Other important notions introduced here are those of matrix ring (Section 4.4) and tensor product (Section 4.8).