A NOTE ON SATURATED MULTIPLICATIVELY CLOSED SETS
In this paper, we introduce and study $ \mathcal{H}_Y $-s.m.c. and strong $ \mathcal{H}_Y $-s.m.c. sets and give some connections between them and lattice ideals of $ \mathcal{H}_Y $. Also, we introduce an ideal $ R_S $, for each subset set $ S $ of a ring $ R $. We prove a ring $ R $ is a Gelfand ring if and only if $ R_S $ is an intersection of maximal ideals, for every s.m.c. set $ S $ of $ R $.
Keywords:
$ \mathcal{H}_Y $-ideal, $\mathcal{H}_Y$-saturated multiplicatively closed set, closed ideal, $C(X)$ maximal ideal,
___
- A. R. Aliabad, M. Badie and S. Nazari, An extension of $z$-ideals and $z^\circ$-ideals, Hacet. J. Math. Stat., 49(1) (2020), 254-272.
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
- F. Azarpanah, M. Ghirati and A. Taherifar, Closed ideals in $C(X)$ with different representations, Houston J. Math., 44(1) (2018), 363-383.
- M. Badie, On $\mathcal{H}_Y$ -ideals, Bull. Iranian Math. Soc., (2020), to appear. (https://doi.org/10.1007/s41980-020-00429-y)
- L. Gillman and M. Jerison, Rings of Continuous Functions, The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960.
- R. Y. Sharp, Steps in Commutative Algebra, Second edition, London Mathematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
- S. A. Steinberg, Lattice-Ordered Rings and Modules, Dordrecht, Springer, 2010.
- S. Willard, General Topology, Addison-Wesley Publishing Co., Reading, New York, 1970.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI