Elementary Properties of Modules

Abstract

One way to study rings is through R-modules, which (in some possibly non-faithful way) serve to “represent” the ring. But such a view is restrictive. With R-modules one enjoys an increased generality. Any property possessed by an R-module can conceivably apply to the ring R itself, as a module over itself. Also, any universal property gains a greater strength, when one enlarges the ambient realm in which the property is stated—in this case from rings R, to their R-modules. This chapter still sticks to basics: homomorphisms, submodules, direct sums and products and free R-modules. Beyond that, chain conditions on the poset of submodules can be seen to have important consequences in two areas: endomorphisms of modules, and their generation. The former yields the Krull-Remak-Schmidt Theorem concerning the uniqueness of direct decompositions into indecomposable submodules, while, for the latter, the ascending chain condition is connected with finite generation via Noether’s Theorem. (The existence of rings of integral elements is derived from this theorem.) The last sections of the chapter introduce exact sequences, projective and injective modules, and mapping properties of \(\mathrm{Hom}(M,N)\)—a hint of the category-theoretic point of view to be unfolded at the beginning of Chap. 13.