Commutative Ring - Modules

Modules

The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules, which are similar to vector spaces, except that the base is not necessarily a field, but can be any ring R. The theory of R-modules is significantly more difficult than linear algebra of vector spaces. Module theory has to grapple with difficulties such as modules not having bases, that the rank of a free module (i.e. the analog of the dimension of vector spaces) may not be well-defined and that submodules of finitely generated modules need not be finitely generated (unless R is Noetherian, see below).

Ideals within a ring R can be characterized as R-modules which are submodules of R. On the one hand, a good understanding of R-modules necessitates enough information about R. Vice versa, however, many techniques in commutative algebra that study the structure of R, by examining its ideals, proceed by studying modules in general.