It is well known that a nonzero nonunit in a Noetherian integral domain factors into irreducibles.
The so-called Nagata criterion holds for an integral domain A satisfying (ACCP): Let S be a multiplicatively closed subset of A generated by prime elements.
[4] The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of principal ideals is non-terminating.
The former only requires the poset of ideals of the form xR to satisfy the ascending chain condition, and the latter only examines the poset of ideals of the form Rx.
Whether the ACCP holds on the right or left side of R, it implies that R has no infinite set of nonzero orthogonal idempotents, and that R is a Dedekind finite ring.