S-Noetherian spectrum condition

ABSTRACT

A commutative ring R with identity satisfies the S-Noetherian spectrum condition, where S⊆R is a given multiplicative set, if for each ideal I of R, $sI\subseteq \sqrt{J}\subseteq \sqrt{I}$ for some s∈S and some finitely generated ideal J. Using this concept, we give an S-version of several different known results. For instance, the ring R satisfies the S-Noetherian spectrum property if and only if the polynomial ring R[X] satisfies the S-Noetherian spectrum property.

KEYWORDS:

• Polynomial rings
• radically S-finite ideals
• ring has S-Noetherian spectrum

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13E99
• 13B25
• 13C

Acknowledgements

The author is grateful to Professor Ali Benhissi for valuable discussions. He also thanks the referee for his/her careful considerations.