ABSTRACT
What are the closure properties of the class of countably-generated left R-modules? Since a homomorphic image of a countably-generated left R-module must be countably-generated, the class of countably-generated left R-modules is closed under factor modules and direct summands. The coproduct of a family of nonzero countably-generated left R-modules will be countably generated if and only if the family is at most countably infinite. Direct sums of arbitrary families of countably-generated left R-modules will be studied in Chapter 12, and the results applied in Chapter 13. Rings differ with regard to the property of submodules of countably-generated left R-modules being necessarily countably generated.
OBSERVATION 1. A ring R has the property that each submodule of each countably-generated left R-module is countably generated if and only if each left ideal of R is countably generated.
