A Family of Quotients of the Rees Algebra

Abstract

A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.MSC: 20M14; 13H10; 13A30.

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…We are interested in the Hilbert function of these rings. In [6,Proposition 2.3] it is stated that their Hilbert functions do not depend on a and b, but actually in the proof it is shown more and then we restate that proposition in the version we need: Proposition 1.1. If (R, m) is a local ring and I is an ideal of R, then for any h ≥ 2…”

“…where a and b are elements of R, and let I 2 (t 2 + at + b) denote its contraction to the Rees algebra. In [6] it is introduced and studied the following family of rings…”

“…One aim of this construction was to provide a unified approach to Nagata's idealization (see [5]) and amalgamated duplication (see [11] and [13]), which are isomorphic to R(I) 0,0 and R(I) −1,0 respectively. In fact in [6] and [7] it is proved that several properties of the family are independent of a and b. We are interested in the Hilbert function of these rings.…”

On $n$-trivial extensions of rings

“…Using [13, Lemma 1.2 and Proposition 1.3], we get, similar to [13,Proposition 1.4], the following diagram of extensions and isomorphisms of rings:…”

Zaks' conjecture on rings with semi-regular proper homomorphic images

“…For more details on trivial ring extensions, we refer the reader to Glaz's book [15] and Huckaba's books [17]. Recent works investigating various ring-theoretic aspects of these constructions are [3,5,21,22,24,28,29].…”