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=Mn(F) is the full matrix ring over a field, and M=Mn 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.
Read more about this topic: Noetherian Module
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