n. a module that satisfies the ascending chain condition so that every strictly increasing chain of submodules is finite. This is equivalent to satisfying the maximum condition that every non-empty set of submodules has a maximal member, and, even if the module is not unitary, to every submodule, including the module itself, being finitely generated. For example, the set Z of integers form a Noetherian Z-module, but the rationals do not. See Artinian module.