In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbreviated to ACCP) is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.
The counterpart descending chain condition may also be applied to these posets, however there is currently no need for the terminology "DCCP" since such rings are already called left or right perfect rings. (See § Noncommutative rings below.)
Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and left or right perfect rings.