Families of Gorenstein and almost Gorenstein rings

TL;DR

This paper investigates a family of rings derived from a base ring and an ideal, showing that Gorenstein and almost Gorenstein properties are independent of certain parameters, with characterizations and formulas provided.

Contribution

It characterizes when rings in the family are Gorenstein or almost Gorenstein, demonstrating independence from parameters and providing explicit formulas for their type.

Findings

Gorenstein property is independent of parameters a,b

Almost Gorenstein property is independent of parameters a,b

Provides formulas for the type of these rings

Abstract

Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this family we find Nagata's idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a,b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.