LetRbe a commutative ring. We say thatRsatisfies theascending chain condition for principal ideals, or thatRhasproperty(M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals ofRterminates. Property (M) is equivalent to themaximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals ofRhas a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for ifRhas property (M) and if {Xλ} is a set of indeterminates overR, then the polynomial ringR[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).
On unique factorization domains
