Title: On right $S$-Noetherian rings and $S$-Noetherian modules,
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.