ABSTRACT
We will begin our considerations of finitely-generated modules by asking the following question. For an arbitrary ring R, what closure properties does the class of finitely-generated left R-modules have? Any factor module of a finitely-generated left R-module must be finitely generated. Consequently, the class of finitely-generated left R-modules is closed under taking factor modules and direct summands. The coproduct of a family of nonzero finitely-generated left R-modules will be finitely generated if and only if the family is finite. In Chapter 11 countably-infinite coproducts of finitely-generated modules will be examined and used. Rings differ with regard to the property of submodules of finitely-generated modules being necessarily finitely generated.
OBSERVATION 1. A ring R has the property that each submodule of each finitely-generated left R-module is finitely generated if and only if R is left noetherian.
VERIFICATION: If each submodule of each 178finitely-generated left R-module is finitely generated then, in particular, each left ideal of R must be finitely generated. Now suppose that R is left noetherian. Then by Observations 1 and 2 of §8.4 we see that each submodule of each finitely-generated left R-module must be finitely-generated.
