Noetherian Module - Characterizations, Properties and Examples

Characterizations, Properties and Examples

In the presence of the axiom of choice, two other characterizations are possible:

• Any nonempty set S of submodules of the module has a maximal element (with respect to set inclusion.) This is known as the maximum condition.
• All of the submodules of the module are finitely generated.

If M is a module and K a submodule, then M is Noetherian if and only if K and M/K are Noetherian. This is in contrast to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated.

Examples

• The integers, considered as a module over the ring of integers, is a Noetherian module.
• If R=M_n(F) is the full matrix ring over a field, and M=M_{n 1}(F) is the set of column vectors over F, then M can be made into a module using matrix multiplication by elements of R on the left of elements of M. This is a Noetherian module.
• Any module that is finite as a set is Noetherian.
• Any finitely generated right module over a right Noetherian ring is a Noetherian module.