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2022•Journal of Algebra and Its ApplicationsRequires access

The Krull dimension-dependent elements of a Noetherian commutative ring

S. Babaei, Esra Şengelen Sevim

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Abstract

In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let [Formula: see text] be non-unit elements of a commutative ring [Formula: see text]. [Formula: see text] are called Krull dimension-dependent elements, whenever [Formula: see text] We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension.

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What this paper is about

In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let [Formula: see text] be non-unit elements of a commutative ring [Formula: see text]. [Formula: see text] are called Krull dimension-dependent elements, whenever [Formula: see text] We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension.

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Available abstract

In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let [Formula: see text] be non-unit elements of a commutative ring [Formula: see text]. [Formula: see text] are called Krull dimension-dependent elements, whenever [Formula: see text] We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension.

Key concepts: Krull dimension, Dimension theory (algebra), Mathematics, Global dimension, Regular local ring, Noetherian, Noetherian ring, Commutative algebra

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