The chain conditions on commutative semigroups can occur on either subalgebras, ideals, or principal ideals. The stongest of these are the chain conditions on subalgebras.
Theorem. let S be a commutative semigroup for which Sub(S) satisfies both the ACC and the DCC then S is finite.
Proof. (1) the ACC on subalgbras means that S is finitely generated, for if it were not then an infinite gneerating set for it would create an infinite ascending chain (2) the DCC on subalgebras means that S is monofinite, meaning that for each x in S the principal subalgebra (x) is finite. If it were not finite, then (x) would generate a (\mathbb{Z}_+,+) semigroup which has an infinite descending chain of subalgebras (3) S is finitely generated, each element generates a finite number of elements, and S is commutative so S is finite. \square
The situation with respect to \mathbb{Z}-modules is a bit different. In that case, there is no distinction between subalgebras, ideals, and principal ideals: there are only submodules. An analogous result shows that \mathbb{Z} modules satisfying both the ACC and DCC on subalgebras are finite. This theorem is in the same vein of the following familiar theorem from commutative algebra:
Proposition. let M be a module satisfying both the ACC and the DCC then M has finite length. [1]
This is one case in which commutativity is necessary. We have that finitely generated commutative semigroups in which each element has a finite principal subalgebra are finite, but the converse does not need to be the case for non-commutative semigroups or groups. It has been shown that there is a finitely generated torsion group that is not finite for example [2].
References:
[1] Commutative algebra volume one
Zariski and Samuel
[2] Burnside problem
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