Almost Gorenstein rings - towards a theory of higher dimension -

TL;DR

This paper extends the concept of almost Gorenstein rings from one-dimensional cases to higher dimensions, developing a new theoretical framework and exploring graded versions.

Contribution

It introduces a higher-dimensional definition of almost Gorenstein rings and develops foundational theory for these structures, including graded variants.

Findings

Proposed a higher-dimensional notion of almost Gorenstein rings.

Developed basic properties and theoretical framework.

Explored graded versions of the higher-dimensional concept.

Abstract

The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fr\"oberg for one-dimensional Noetherian local rings which are analytically unramified has been generalized by S. Goto, N. Matsuoka and T. T. Phuong to one-dimensional Cohen-Macaulay local rings, possessing canonical ideals. The present purpose is to propose a higher-dimensional notion and develop the basic theory. The graded version is also posed and explored.