Finitely-generated Module - Equivalent Definitions and Finitely Cogenerated Modules

Equivalent Definitions and Finitely Cogenerated Modules

The following conditions are equivalent to M being finitely generated (f.g.):

• For any family of submodules in M, if, then for some finite subset F of I.

• For any chain of submodules in M, if, then for some i in I.

• If is an epimorphism, then the restriction is an epimorphism for some finite subset F of I.

From these conditions it is easy to see that being finitely generated is a property preserved by Morita equivalence. The conditions are also convenient to define a dual notion of a finitely cogenerated module M. The following conditions are equivalent to a module being finitely cogenerated (f.cog.):

• For any family of submodules in M, if, then for some finite subset F of I.

• For any chain of submodules in M, if, then for some i in I.

• If is a monomorphism, then is a monomorphism for some finite subset F of I.

Both f.g. modules and f.cog. modules have interesting relationships to Noetherian and Artinian modules, and the Jacobson radical J(M) and socle soc(M) of a module. The following facts illustrate the duality between the two conditions. For a module M:

• M is Noetherian if and only if every submodule of N of M is f.g.
• M is Artinian if and only if every quotient module M/N is f.cog.
• M is f.g. if and only if J(M) is a superfluous submodule of M, and M/J(M) is f.g.
• M is f.cog. if and only if soc(M) is an essential submodule of M, and soc(M) is f.g.
• If M is a semisimple module (such as soc(N) for any module N), it is f.g. if and only if f.cog.
• If M is f.g. and nonzero, then M has a maximal submodule and any quotient module M/N is f.g.
• If M is f.cog. and nonzero, then M has a minimal submodule, and any submodule N of M is f.cog.
• If N and M/N are f.g. then so is M. The same is true if "f.g." is replaced with "f.cog."

Finitely cogenerated modules must have finite uniform dimension. This is easily seen by applying the characterization using the finitely generated essential socle. Somewhat asymmetrically, finitely generated modules do not necessarily have finite uniform dimension. For example, an infinite direct product of nonzero rings is a finitely generated (cyclic!) module over itself, however it clearly contains an infinite direct sum of nonzero submodules. Finitely generated modules do not necessarily have finite co-uniform dimension either: any ring R with unity such that R/J(R) is not a semisimple ring is a counterexample.