A note on finitely generated ideals which are locally principal

Let R be a commutative ring with identity 1#0 and let A be a nonzero ideal of R. A problem of current interest is to relate the notions of "projective ideal", "flat ideal" and "multiplication ideal". In this note we prove two results which show that the maximal ideals containing the annihilator of A can play an important role in determining the relationship between these concepts. As a consequence we are able to prove that a finitely generated multiplication ideal in a semi-quasi-local ring is principal, that a finitely generated flat ideal having only a finite number of minimal prime divisors is projective and that for Noetherian rings or semihereditary rings, finitely generated multiplication ideals with zero annihilator are invertible. Our notation is essentially that of [4]. In particular, an ideal A of R is said to be a multiplication ideal if whenever B is an ideal of R with BC A, there exists an ideal C of R such that B=AC. There is one deviation from the notation of [4] and that is that we shall denote by A ' the aninihilator of an ideal A. THEOREM 1. Let A be a finitely generated multiplication ideal of R. If A' is contained in onlyfinitely many maximal ideals, then A is principal. PROOF. Let M1, *, Mn be the maximal ideals of R containing A'. For i between 1 and n, if 1l;lji MJA MiA, then MiARMi2 (Tl==1.ji MjA)RM.=ARM.. Since ARM, is finitely generated and since MiRMi is the Jacobson radical of Rm,, it follows from the Nakayama Lemma that ARM.=(O). Thus, ALRM,=(ARM.)'=RM., which contradicts the fact that A'-M. Therefore, for 1 , then we also have that M? (aR: A) since aoAM. Received by the editors March 15, 1971. AMS 1969 subject classifications. Primary 1320; Secondary 1340.