On right S-Noetherian rings and S-Noetherian modules

ABSTRACT

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.

KEYWORDS:

• Completely prime right ideals
• Oka families of right ideals
• point annihilator sets
• right S-Noetherian rings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D25
• 16D80

Acknowledgment

We thank the anonymous referee for several helpful comments and suggestions. We also thank Omid Khani Nasab for valuable discussions, which led in particular to an improvement of Theorem 2.3.