Exercise 7, § 1

a) Let A be a ring and $\mathscr{I}$ the set of left ideals of A of the form $Ae$, where $e$ is an idempotent in A. Prove that the following properties are equivalent:

(i) Every nonempty subset of $\mathscr{I}$ has a minimal element.

(ii) Every nonempty subset of $\mathscr{I}$ has a maximal element.

(iii) There is no infinite family of nonzero ideals belonging to $\mathscr{I}$ whose sum is direct.

(iv) There is no infinite family $(e_i)$ of nonzero idempotents in A such that $e_ie_j= 0$ for $i\not=j$.

b) Give an example of a ring A in which every nonempty set of monogenous left ideals has a maximal element and a minimal element (so that the equivalent conditions of a) are satisfied) even though A is neither left Artinian nor left Noetherian (use Exercise 3).

c) Let M be a module. If every nonempty set of finitely generated submodules of M has a maximal element, then M is Noetherian.

d) Give an example of a non-Artinian module in which every nonempty set of finitely generated submodules has a minimal element.