Definition 1

Let A be a ring. We call an A-module M Artinian (resp. Noetherian) if it satisfies the following equivalent conditions:

(i) Every nonempty set of submodules of M, ordered by inclusion, has a minimal (resp. maximal) element.

(ii) Every decreasing (resp. increasing) sequence of submodules of M is stationary.

The equivalence of conditions (i) and (ii) follows from Set Theory, III, §6, No. 5, p. 190, Proposition 6.

An A-module M is Artinian (resp. Noetherian) if and only if M, viewed as a module over the ring of homotheties $A_M$, is Artinian (resp. Noetherian).

Let M be an Artinian (resp. Noetherian) A-module. Every nonempty set of submodules of M, ordered by inclusion, that is left directed (resp. right directed) has a least element (resp. a greatest element) (Set Theory, III, §1, No. 10, p. 145, Proposition 10).

Let M be an Artinian (resp. Noetherian) A-module and $(M_i)_{i\in I}$ a family of submodules of M. The intersections (resp. sums) of finite subfamilies of the family $(M_i)_{i\in I}$ form a nonempty left (resp. right) directed set of submodules of M. Therefore, there exists a finite subset J of I such that $\bigcap_{i\in I}M_i=\bigcap_{i\in J}M_i$ (resp. $\sum_{i\in I}M_i=\sum_{i\in J}M_i$).