Generalization of Artinian rings and the formal power series rings

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 ann(M)∩S=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {ann}(M) \cap S = \emptyset$$\end{document} and if for every descending chain of submodules N1⊇N2⊇⋯⊇Nn⊇⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_1 \supseteq N_2 \supseteq \dots \supseteq N_n \supseteq \cdots$$\end{document} of M, there exist s∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in S$$\end{document} and n0∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_0 \in {\mathbb {N}}$$\end{document} such that sNn0⊆Nn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$sN_{n_0} \subseteq N_n$$\end{document} for all n≥1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 1.$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(+)M$$\end{document} to be an S(+)M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(+)M$$\end{document}-Artinian ring.