20024

An extension of $S$--noetherian rings and modules

For any commutative ring $A$ we introduce a generalization of $S$--noetherian rings using a here\-ditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that totally noetherian w.r.t. $\sigma$ is a local property, and if $A$ is a totally noetherian ring w.r.t $\sigma$, then $\sigma$ is of finite type.

Keywords:

Noetherian ring and module, polynomial ring formal power series ring,

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H. Ahmed, $S$-Noetherian spectrum condition, Comm. Algebra, 46(8) (2018), 3314-3321.
D. D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra, 30(9) (2002), 4407-4416.
T.~Dumitrescu, Generic fiber of power series ring extensions, Comm. Algebra, 37(3) (2009), 1098-1103.
M. Eljeri, $S$-strongly finite type rings, Asian Research J. Math., 9(4) (2018), 1-9.
J. S. Golan, Torsion Theories, Pitman Monographs and Surveys in Pure and Applied Math., 29, Pitman, 1986.
E. Hamann, E. Houston and J. L. Johnson, Properties of uppers to zero in $R[X]$, Pacific J. Math., 135(1) (1988), 65-79.
P. Jara, An extension of $S$-artinian rings and modules to a hereditary torsion theory setting, Comm. Algebra, 49(4) (2021), 1583-1599.
A. V. Jategaonkar, Endomorphism rings of torsionless modules, Trans. Amer. Math. Soc., 161 (1971), 457-466.
C. Jayaram, K. H. Oral and U. Tekir, Strongly 0-dimensional rings, Comm. Algebra, 41(6) (2013), 2026-2032.
P. Jothilingam, Cohen's theorem and Eakin-Nagata theorem revisited, Comm. Algebra, 28(10) (2000), 4861-4866.
J. W. Lim, A note on $S$-Noetherian domains, Kyungpook Math. J., 55(3) (2015), 507-514.
J. W. Lim and D. Y. Oh, $S$-Noetherian properties on amalgamated algebra along an ideal, J. Pure Appl. Algebra, 218(6) (2014), 1075-1080.
Z. Liu, On $S$-Noetherian rings, Arch. Math. (Brno), 43(1) (2007), 55-60.
E. S. Sevim, U. Tekir and S. Koc, $S$-artinian rings and finitely $S$-cogenerated rings, J. Algebra Appl., 19(3) (2020), 2050051 (16 pp).
B. Stenström, Rings of Quotients, Springer-Verlag, Berlin, 1975.