Stable rings

Ring-Theoretic Properties of PvMDs

“…Motivated by earlier work of H. Bass [2] an J. Lipman [32] on the number of generators of an ideal, in 1974 J. Sally and W. Vasconcelos defined a Noetherian ring R to be stable if each nonzero ideal of R is projective over its endomorphism ring End R (I) [41,42]. When I is a nonzero ideal of a domain R, then End R (I) = (I : I); thus a domain R is stable if each nonzero ideal I of R is invertible in the overring (I : I).…”

The ratliff-rush ideals in a noetherian ring

“…By induction on the minimal number of generators of I, we can find a two-generated ideal J contained in I that has no principal reduction. Then J 2 is also two-generated by Proposition 3.2, so by the proof of [SV2,Theorem 3.4], J is stable, a contradiction.…”

“…The embedding dimension of R is the minimal number of generators of its maximal ideal M , while the multiplicity of R is the minimal number of generators of high powers of M . [SV2,Example 5.4]. In their example R has multiplicity three, and the completion of R has nonzero nilpotent elements (which is always the case if R is not finitely generated over R).…”

The minimal number of generators of an invertible ideal

“…A Dedekind domain is characterized as an integrally closed Noetherian domain of Krull dimension 1. It turns out that of these three properties, Krull dimension 1 always implies that an invertible ideal is 2-generated, as was shown by Sally and Vasconcelos in [SV74]. R. Heitmann generalized this fact to arbitrary finite Krull dimension: an invertible ideal of an n-dimensional domain R is strongly n + 1-generated (see Section 3).…”

“…The bound n + 1 is sharp. The case n = 1 was proved by Sally and Vasconcelos in [SV74]. The general case is due to Heitmann:…”