Atomic rings and the ascending chain condition for principal ideals

Abstract

Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals , or that R has property ( M ), if each ascending sequence ( a 1 ) ⊆ ( a 2 ) ⊆ … of principal ideals of R terminates. Property ( M ) is equivalent to the maximum condition on principal ideals ; that is, under the partial order of set containment, each collection of principal ideals of R has a maximum element. Noetherian rings, of course, have property ( M ), but the converse is not true; for if R has property ( M ) and if { X λ } is a set of indeterminates over R , then the polynomial ring R [{ X λ }] has property ( M ). Krull domains, and hence unique factorization domains, have property ( M ).