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2011•International Journal of Mathematical ArchiveRequires access

• NONNIL-NOETHERIAN COMMUTATIVE RINGS AND NONNIL IDEALS

Parisa M. Shabestari, Sh. Safarisabet

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Abstract

In this paper, it should be noted that a commutative ring R is said to be Nonnil-Noetherian if every Nonnil Ideal of R is finitely generated. A ring R is said to be Nonnil-Noetherian if R is both right and left Nonnil-Noetherian ring. In the first section of this paper we show that many of the properties Noetherian rings are also true for Nonnil-Noetherian rings. In the second section we show that the direct sum of Nonnil-Noetherian ring needs not to be Noetherian.

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What this paper is about

In this paper, it should be noted that a commutative ring R is said to be Nonnil-Noetherian if every Nonnil Ideal of R is finitely generated. A ring R is said to be Nonnil-Noetherian if R is both right and left Nonnil-Noetherian ring. In the first section of this paper we show that many of the properties Noetherian rings are also true for Nonnil-Noetherian rings. In the second section we show that the direct sum of Nonnil-Noetherian ring needs not to be Noetherian.

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Available abstract

In this paper, it should be noted that a commutative ring R is said to be Nonnil-Noetherian if every Nonnil Ideal of R is finitely generated. A ring R is said to be Nonnil-Noetherian if R is both right and left Nonnil-Noetherian ring. In the first section of this paper we show that many of the properties Noetherian rings are also true for Nonnil-Noetherian rings. In the second section we show that the direct sum of Nonnil-Noetherian ring needs not to be Noetherian.

Key concepts: Noetherian, Mathematics, Radical of a ring, Pure mathematics, Noetherian ring, Hilbert's basis theorem, Commutative property, Ring (chemistry)

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