Abstract
Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.
Highlights
The concept of Noetherian rings is one of the most important topics that is widely used in many areas including commutative algebra and algebraic geometry
Let R be a commutative ring with identity and let S be a multiplicative subset of R
We study some properties of power series rings over nonnil-S-Noetherian rings
Summary
The concept of Noetherian rings is one of the most important topics that is widely used in many areas including commutative algebra and algebraic geometry. The Noetherian property was originally due to the mathematician Noether who first considered a relation between the ascending chain condition on ideals and the finitely generatedness of ideals She showed that if R is a commutative ring with identity, the ascending chain condition on ideals of R holds if and only if every ideal of R is finitely generated. Recall that an ideal I of R is nonnil if I is not contained in Nil( R); and R is a nonnil-Noetherian ring if each nonnil ideal of R is finitely generated. If S consists of units of R, the concept of S-finite ideals is the same as that of finitely generated ideals; so if S consists of units of R, the notion of nonnil-S-Noetherian rings coincides with that of nonnil-Noetherian ring.
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