Bounding the number of generators for a class of ideals in local rings

Christian Gottlieb
1995 Communications in Algebra  
A well-known property of local (Noetherian) rings of dimension one is that there is, for each such ring, a number which uniformly bounds the number of generators of any ideal in the ring. A proof may be found in [7] (Theorem 1.2 of Chapter 3). In this paper we use the notation µ(I) for the minimal number of elements needed to generate an ideal I in a local ring R. Thus µ(I) = (I/mI), where m is the maximal ideal in R. In [2] we proved that, when R is one-dimensional, we actually have µ(I) ≤
more » ... x)), where x is any element in m. Thus µ(I) is bounded by the least colength of a nonunit in R and this number is of course finite since we may choose x such that (x) is m-primary. However, it is obvious from the theory of Hilbert functions that there is no uniform bound on µ(I) if dim R > 1 because then µ(m n ) is, for large n, a nonconstant polynomial. To find uniform bounds in rings of dimension higher than one we must thus restrict ourselves to appropriate subclasses of the class of all ideals in the ring. We find a result in this spirit in [7] (Theorem 2.1 of Chapter 3), namely that dim R ≤ 2 if and only if there is a uniform bound on µ(I) for all I such that m is not an associated prime ideal of I. To prove the "if-part" is quite straightforward. Indeed, if dim R > 2, let p be a prime ideal of height 2 and use the facts that each power of the maximal ideal p p of R p is the extension of a p-primary ideal of R and that µ(I) ≥ µ(I p ) for all I.
doi:10.1080/00927879508825289 fatcat:rokq7zukyrflrhoezfkxy37wwe