Hilbert regularity of Stanley–Reisner rings

Hilbert regularity of Stanley-Reisner rings

On Regularity of Rings

It is shown that a ring R is (von Neumann) regular if and only if every cyclic left R-module is GP -injective if and only if R is left PP and left GP -injective. In addition, two examples are given to answer two open questions in the negative. 2000 Mathematics Subject Classification: 16E50, 16D60

Commutative Monoid Rings as Hilbert Rings

Let S be a cancellative monoid with quotient group of torsion-free rank a. We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a. Assume that R is a commutative unitary ring and G is an abelian group. The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the gr...

Noncommutative Extensions of Hilbert Rings

Hilbert Coefficients and Depths of Form Rings

We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e1. As an application, we derive bounds on the higher Hilbert coeffi...