Generalization of Artinian rings and the formal power series rings

Abstract

Let R be a commutative ring, S a multiplicative subset of R and M an R-module. We say that M is S-Artinian, if $\text {ann}(M) \cap S = \emptyset$ and if for every descending chain of submodules $N_1 \supseteq N_2 \supseteq \dots \supseteq N_n \supseteq \cdots$ of M, there exist $s \in S$ and $n_0 \in {\mathbb {N}}$ such that $sN_{n_0} \subseteq N_n$ for all $n \ge 1.$ The ring R is said to be S-Artinian if it is S-Artinian as an R-module. In this paper, we study the S-Artinian property and we show that the class of S-Artinian integral domains is a subclass of the class of anti-Archimedean domains. We show that the S-Artinian domains are exactly the domains exhibiting a smooth behavior for the quotient field of their formal power series rings. We also, give a necessary and sufficient condition for the idealization $R(+)M$ to be an $S(+)M$-Artinian ring.