JOURNAL OF ALGEBRA 14, 139-151 (1970) On the Number of Generators of an Invertible Ideal ROBERT GILMER Dept. of Mathematics, Florida State University,lTallahassee, Florida 32306 AND WILLIAM HEINZER Division of Mathematical Sciences, Purdue University, Lafayette, Indiana 47907 Communicated by D. Rees Received December 6, 1968 Let R be a commutative ring containing a regular elementand let T be the total quotient ring of R. If F is an invertible ideal of R, then F has a finite basis;this result was originally proved by Krull in [12] for the case when R is an integral domain with identity, and Krull’s proof generalizes to the case of a ring containing a regular element. In the classicalcasewhen R is a Dedekind domain, F hasa basis of two elements, one of which can be chosen arbitrarily from the set of nonzero elementsof F [lo]. However, S. Chase has given exampleswhich show that for any positive integer n 2 3, there exists a Noetherian integral domain Jn with identity containing an invertible ideal with n, but no fewer, generators. In Section 1 we show that either of the following conditions is sufficient in order that an invertible ideal A of a commutative ring R with identity have a basis of two elements:(1) A is principal over A2. (2) A 3 ( uh AM,), where {MA} is the set of maximal ideals of R containing A. Also, there is a brief consideration in Section 1 when one of a set of two generatorsfor A can be chosen arbitrarily from the set of nonzero elementsof A. In Section 2, we consider invertible ideals of a Priifer domain D. Several known results concerning Prtifer domains indicate plausibility of the conjecture that each finitely generated ideal of D has a basisof two elements.We do not resolve this conjecture in Section 2, but we do show that conditions (1) and (2) above are equivalent, even in the local case,in D, and we give additional sufficient conditions, in terms of valuation ideals,in order that a fixed finitely generated ideal of D have a basisof two elements.It is clear, of course, that Chase’s domains J,, are not Prtifer. In Section 3 we indicate a general construction of 139