The effect of $S$-accr on intermediate rings between certain pairs of rings

Yıl 2022, Cilt: 32 Sayı: 32, 101 - 128, 16.07.2022

Subramanian Vısweswaran

https://doi.org/10.24330/ieja.1096895

Öz

The rings considered in this article are commutative with identity and the modules are assumed to be unitary. If $R$ is a subring of a ring $T$, then it is assumed that $R$ contains the identity element of $T$. Let $S$ be a multiplicatively closed subset (m.c. subset) of a ring $R$. In this paper, we consider the concept of $S$-accr, the generalization by Hamed and Hizem of the notion of (accr) in module theory given by Lu. We say that $R$ satisfies (accr) if the increasing sequence of residuals of the form $(I:_{R}B)\subseteq (I:_{R}B^{2})\subseteq (I:_{R}B^{3})\subseteq \cdots$ is stationary for any ideal $I$ of $R$ and for any finitely generated ideal $B$ of $R$. Focusing on certain pairs of rings $R\subseteq T$, the aim of this paper is to study whether $S$-accr on each intermediate ring $A$ between $R$ and $T$ for a suitable m.c. subset $S$ of $A$ (depending on $A$) implies that $A$ satisfies (accr) for each such $A$.

Anahtar Kelimeler

Accr pair, $S$-accr pair, Laskerian pair, strongly Laskerian pair, $S$-Noetherian pair

Kaynakça

• D. D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra, 30(9) (2002), 4407-4416.
• M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969.
• N. Bourbaki, Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1972.
• R. Gilmer, Integral dependence in power series rings, J. Algebra, 11 (1969), 488-502.
• R. Gilmer and W. Heinzer, Finitely generated intermediate rings, J. Pure Appl. Algebra, 37(3) (1985), 237-264.
• A. Hamed and S. Hizem, Modules satisfying $S$-Noetherian property and $S$-ACCR, Comm. Algebra, 44(5) (2016), 1941-1951.
• A. Hamed and A. Malek, $S$-prime ideals of a commutative ring, Beitr. Algebra Geom., 61(3) (2020), 533-542.
• W. Heinzer and D. Lantz, The Laskerian property in commutative rings, J. Algebra, 72(1) (1981), 101-114.
• I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
• C. P. Lu, Modules satisfying ACC on a certain type of colons, Pacific J. Math., 131(2) (1988), 303-318.
• C. P. Lu, Modules and rings satisfying (accr), Proc. Amer. Math. Soc., 117(1) (1993), 5-10.
• N. H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly, 49 (1942), 286-295.
• N. Radu, Sur les Anneaux Laskeriens, in: Proceedings of the Week of Algebraic Geometry, Bucharest, (1980), 158-163.
• S. Visweswaran, Laskerian pairs, J. Pure Appl. Algebra, 59(1) (1989), 87-110.
• S. Visweswaran, ACCR pairs, J. Pure Appl. Algebra, 81(3) (1992), 313-334.
• S. Visweswaran, Some results on $S$-primary ideals of a commutative ring, Beitr Algebra Geom., (2021) https://doi.org/10.1007/s13366-021-00580-5.
• S. Visweswaran and P. T. Lalchandani, Some results on modules satisfying $S$-strong accr$^{*}$, Arab J. Math. Sci., 25(2) (2019), 145-155.
• A. R. Wadsworth, Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc., 195 (1974), 201-211.