A short note on the Marot property in rings of continuous functions

Document Type : Original Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Abstract

Let $X=Y\cup \left\{\omega\right\}$ where $\omega \notin Y$, topologized by equipping $Y$ with the discrete topology, and by letting deleted neighborhoods of $\omega$ consist of complements of closed discrete subsets of $Y$ in its Riemann surface topology. Assume that $I$ is an ideal of $C^{*}(X)$ where $C^{*}(X)$ is the ring of all bounded real-valued continuous functions on $X$. A result of Adler and Williams showed that $I$ contains a regular element if and only if a set of regular elements generates $I$. In this note, we obtain some conditions on $X$ for which the rings of continuous functions on $X$ are Marot. Moreover, this paper gives a sufficient condition for a quasi-B'ezout ring to be additively regular.

Keywords

Main Subjects

References

[1] A. Adler and R.D. Williams, Transferring results from rings of continuous functions to rings of analytic functions, Canadian J. Math. 27 (1975) 75-87.
[2] F. Azarpanah, F. Farokhpay and E. Ghashghaei, Annihilator-stability and unique generation in C(X), J. Algebra Appl. 18 (2019) 1950122, 16 pp.
[3] F. Azarpanah, O.A.S. Karamzadeh and A. Rezai Aliabad, On z◦-ideals in C(X), Fund. Math. 160 (1999) 15-25.
[4] F. Azarpanah and A.R. Salehi, Ideal structure of the classical ring of quotients of C(X), Topology Appl. 209 (2016) 170-180.
[5] F. Dashiell, A. Hager and M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions, Can. J. Math. 32 (1980) 657-685.
[6] L. Gillman and M. Jerison, Rings of Continuous Functions, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960.
[7] R. Gilmer and J. Huckaba, Δ-Rings, J. Algebra 28 (1974) 414-432.
[8] M. Henriksen and R.G. Woods, Cozero complemented spaces; when the space of minimal prime ideals of a C(X) is compact, Topology Appl. 141 (2004) 147-170.
[9] J.A. Huckaba, Commutative Rings with Zero divisors, Monographs and Text-books in Pure and Applied Mathematics 117, Marcel Dekker, Inc., New York, 1988.
[10] M. L. Knox, R. Levy, W. Wm. McGovern and J. Shapiro, Generalizations of complemented rings with applications to rings of functions, J. Alg. Appl. 8 (2009) 17-40.
[11] R. Levy, Almost-P-spaces, Canadian J. Math. 29 (1977) 284-288.
[12] T.G. Lucas, Weakly additively regular rings and special families of prime ideals, Palest. J. Math. 7 (2018) 14-31.
[13] J. Marot, Extension de la notion d’anneau valuation, Dept. Math. Faculte des Sci. de Brest (1968), 46 pp. et 39 pp. de complements.
[14] J. Martinez and S. Woodward, Bézout and Prüfer f-rings, Comm. Algebra 20 (1992) 2975-2989.
[15] R. Matsuda, On Marot rings, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984) 134-137.
[16] D. Portelli and W. Spangher, Krull rings with zero divisors, Comm. Algebra 11 (1983) 1817-1851.
[17] B.V. Zabavsky, Type conditions of stable range for identification of qualitative generalized classes of rings, Algebra Discrete Math. 26 (2018) 144-152.

Files

History

  • Receive Date: 25 September 2022
  • Revise Date: 27 April 2023
  • Accept Date: 03 May 2023

Share

How to cite

Statistics