# On right $S$-Noetherian rings and $S$-Noetherian modules

## TL;DR

This paper extends the concept of $S$-Noetherian rings and modules from commutative to noncommutative algebra, providing characterizations and existence results related to prime ideals and modules.

## Contribution

It introduces new characterizations of right $S$-Noetherian rings via prime ideals and proves an existence theorem for prime annihilators in $S$-Noetherian modules.

## Key findings

- Characterizations of right $S$-Noetherian rings using prime ideals
- Existence of prime point annihilators in $S$-Noetherian modules
- In commutative algebra, modules with certain $S$-Noetherian properties have associated primes.

## Abstract

In this paper we study right $S$-Noetherian rings and modules, extending of 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.