descending chain condition,
n. the condition on submodules that no descending chain
M1 ⊇ M2 ⊇ M3 ⊇ ...
has more than a finite number of distinct members; that is, for every such chain there is an n such that Mn = Mm for all m ≥ n. Equivalently, every non-empty set of submodules has a minimal element. Similar conditions are defined for rings, groups, etc. See also Artinian module. Compare ascending chain condition, minimum condition.
M1 ⊇ M2 ⊇ M3 ⊇ ...
has more than a finite number of distinct members; that is, for every such chain there is an n such that Mn = Mm for all m ≥ n. Equivalently, every non-empty set of submodules has a minimal element. Similar conditions are defined for rings, groups, etc. See also Artinian module. Compare ascending chain condition, minimum condition.