One-dimensional stable rings

Abstract

A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of multiplicity at most $2$, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations. We characterize one-dimensional stable local rings in several ways. The characterizations involve the integral closure $\bar{R}$ of $R$ and the completion of $R$ in a relevant ideal-adic topology. For example, we show: If $R$ is a reduced stable ring, then there are exactly two possibilities for $R$: (1) $R$ is a {\it Bass ring}, that is, $R$ is a reduced Noetherian local ring such that $\bar{R}$ is finitely generated over $R$ and every ideal of $R$ is generated by two elements; or (2) $R$ is a {\it bad stable domain}, that is, $R$ is a one-dimensional stable local domain such that $\bar{R}$ is not a finitely generated $R$-module.

to appear in Journal of Algebra, 31 pages