ABSTRACT
I f Σ is the set o f submodules of a module M , ordered by the relation £ , then i) is called the ascending chain condition (a.c.c. for short) and ii) the maximal condition. A module M satisfying either o f these equivalent conditions is said to be Noetherian (after Emmy Noether). I f Σ is ordered by 2 , then i) is the descending chain condition (d.c.c. for short) and ii) the minimal condition. A module M satisfying these is said to be Artinian (after Emil Artin).
