Some remarks on $S$-Noetherian modules

UDC 512.5 We study several properties and applications of the $S$-Noetherian rings and modules. It is proved that an $S$-Artinian ring is $S$-Noetherian provided that $S$ contains no zero divisors of the module. Furthermore, it is shown that associated primes exist in modules over the $S$-Noetherian rings and the major part of notions of associated prime ideals coincide over the $S$-Noetherian rings. We also extend the classical Krull's intersection theorem for $S$-Noetherian rings. Moreover, we provide a characterization of the $S$-Noetherian modules in terms of the $G$-graded $S$-Noetherian modules.