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# A Generalization of Integrality

Published:2010-08-03
Printed: Dec 2010
• Jim Coykendall,
Department of Mathematics, North Dakota State University, Fargo, ND, U.S.A.
• Tridib Dutta,
Department of Mathematics, North Dakota State University, Fargo, ND, U.S.A.
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## Abstract

In this paper, we explore a generalization of the notion of integrality. In particular, we study a near-integrality condition that is intermediate between the concepts of integral and almost integral. This property (referred to as the $\Omega$-almost integral property) is a representative independent specialization of the standard notion of almost integrality. Some of the properties of this generalization are explored in this paper, and these properties are compared with the notion of pseudo-integrality introduced by Anderson, Houston, and Zafrullah. Additionally, it is shown that the $\Omega$-almost integral property serves to characterize the survival/lying over pairs of Dobbs and Coykendall
 Keywords: integral closure, complete integral closure
 MSC Classifications: 13B22 - Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13G05 - Integral domains 13B21 - Integral dependence; going up, going down