The Chern Coefficient and Cohen-Macaulay rings

TL;DR

This paper explores the relationship between the index of reducibility, the Chern coefficient, and Cohen-Macaulay rings, providing characterizations of Cohen-Macaulay and Gorenstein rings based on these invariants.

Contribution

It offers a new characterization of Cohen-Macaulay rings using the index of reducibility, Cohen-Macaulay type, and Chern coefficient, and characterizes Gorenstein rings via the Chern coefficient.

Findings

Characterization of Cohen-Macaulay rings using the index of reducibility and Chern coefficient.

Characterization of Gorenstein rings in terms of the Chern coefficient.

Main theorem linking these invariants to ring properties.

Abstract

The purpose of this paper is to investigate a relationship between the index of reducibility and the Chern coefficient for primary ideals. Therefore, the main result of this paper gives a characterization of a Cohen-Macaulay ring in terms of its the index of reducibility, its Cohen-Macaulay type, and the Chern coefficient for parameter ideals. As corollaries to the main theorem we obtained the characterizations of a Gorenstein ring in term of its Chern coefficient for parameter ideals.