The Artin Wedderburn theorem generalises this result, with the ascending chain condition.
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In this case " R " is also said to satisfy the ascending chain condition.
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An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.
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The same equivalence holds for infinite lattices which satisfy the ascending chain condition ( or the descending chain condition ).
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A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition.
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Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition.
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The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely generated.
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A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps.
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Note that graded posets need not satisfy the ascending chain condition ( ACC ) : for instance, the natural numbers contain the infinite ascending chain 0.
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A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps.
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