7 - Modules over Semiprime Goldie Rings

Summary

Many investigations of the structure of a right module A over a right noetherian ring R involve related modules over prime or semiprime factor rings of R. For instance, if A is finitely generated, then by using a prime series we may view A as built from a chain of subfactors each of which is a fully faithful module over a prime factor ring of R (see Proposition 3.13). Alternatively, we may relate the structure of A to the structure of the (R/N)-modules A/AN, AN/AN², …, where N is the prime radical of R. Thus, we need a good grasp of the structure of modules over prime or semiprime noetherian rings. The fundamentals of such structure can be obtained with little extra effort for modules over prime or semiprime Goldie rings.

• MINIMAL PRIME IDEALS •

In working with the right Goldie quotient ring of a semiprime factor ring of a right noetherian ring R, say R/I, we shall often need to refer to the regular elements of R/I. It is then convenient to have a notation for the representatives of these cosets, as follows.

Definition. Let I be an ideal in a ring R. An element x ∈ R is said to be regular modulo I provided the coset x + I is a regular element of the ring R/I.