Chain conditions on commutative semigroups

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