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TOWARDS A THEORY OF GORENSTEIN M-PRIMARY INTEGRALLY CLOSED IDEALS

Haya Saka, Satce Kasuga

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Abstract

Let A be a Noetherian local ring with the ma~imlll ideal m and d = dim A. The sel ~ of Gorenstein m.primary integrally closed ideals in A is explored in this paper. If k = A I m is alge­ braically closed and d 2': 2, then X i s infinite. In contrllSt, fo each fiel d k which is no\aigebraically closed and for CIIch integer d 2': 0, there exists a Noetherian complete equi-charactc:ristic local integral domain A with dim A = d such that (I ) the Donnalization of A is regular, (2) !t = {m}. and (3) k = A/ m. When d = 1,.2A is finite ifand onlyifA! p is nota DVR fo any p E Min A, where A deootes the m-aroe completion. The list of clements in r is given, when A is a one-dimensional Noetherian complete local integral domain.

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What this paper is about

Let A be a Noetherian local ring with the ma~imlll ideal m and d = dim A. The sel ~ of Gorenstein m.primary integrally closed ideals in A is explored in this paper. If k = A I m is alge­ braically closed and d 2': 2, then X i s infinite. In contrllSt, fo each fiel d k which is no\aigebraically closed and for CIIch integer d 2': 0, there exists a Noetherian complete equi-charactc:ristic local integral domain A with dim A = d such that (I ) the Donnalization of A is regular, (2) !t = {m}. and (3) k = A/ m. When d = 1,.2A is finite ifand onlyifA! p is nota DVR fo any p E Min A, where A deootes the m-aroe completion. The list of clements in r is given, when A is a one-dimensional Noetherian complete local integral domain.

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Available abstract

Let A be a Noetherian local ring with the ma~imlll ideal m and d = dim A. The sel ~ of Gorenstein m.primary integrally closed ideals in A is explored in this paper. If k = A I m is alge­ braically closed and d 2': 2, then X i s infinite. In contrllSt, fo each fiel d k which is no\aigebraically closed and for CIIch integer d 2': 0, there exists a Noetherian complete equi-charactc:ristic local integral domain A with dim A = d such that (I ) the Donnalization of A is regular, (2) !t = {m}. and (3) k = A/ m. When d = 1,.2A is finite ifand onlyifA! p is nota DVR fo any p E Min A, where A deootes the m-aroe completion. The list of clements in r is given, when A is a one-dimensional Noetherian complete local integral domain.

Key concepts: Integrally closed, Noetherian, Mathematics, Integral domain, Local ring, Ideal (ethics), Pure mathematics, Domain (mathematical analysis)

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