Document Type : Research Paper

**Authors**

Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

**Abstract**

In this article we consider some relations between the topological properties of the spaces X and Min(C_{c} (X)) with algebraic properties of C_{c} (X). We observe that the compactness of Min(C_{c} (X)) is equivalent to the von-Neumann regularity of q_{c} (X), the classical ring of quotients of C_{c} (X). Furthermore, we show that if π is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of πΆ(π) is a minimal prime ideal of C_{c}(X) and in this case πππ(πΆ(π)) and Min(C_{c} (X)) are homeomorphic spaces. We also observe that if π is an F_{c}-space, then Min(C_{c} (X)) is compact if and only if π is countably basically disconnected if and only if Min(C_{c}(X)) is homeomorphic with β_{0}X. Finally, by introducing z^{o}_{c}-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(C_{c} (X)) becomes compact, basically disconnected and extremally disconnected.

**Keywords**

[1] Azarpanah, F., On almost P-spaces, Far East J. Math. Sci., Special Volume, Part I (2000), 121-132.

[2] Azarpanah, F. and Karamzadeh, O.A.S., Algebraic characterizations of some disconnected spaces, Italian J. Pure Appl. Math. 12 (2002), 155-168.

[3] Azarpanah, F., Karamzadeh, O.A.S., and Rezai Aliabad, A., On z^{0}-ideals in πΆ(π), Fund. Math. 160 (1999), 15-25.

[4] Azarpanah, F., Karamzadeh, O.A.S., and Rezai Aliabad, A., On ideals consisting entirely of zero divisors, Comm. Algebra (28) (2000), 1061-1073.

[5] Azarpanah, F., Karamzadeh, O.A.S., Keshtkar, Z., and Olfati, A.R., On maximal ideals of C_{c}(X) and the uniformity of its localizations, Rocky Mountain J. Math. 48(2) (2018), 1-42.

[6] Bhattacharjee, P., Knox, M.L., and Mc Govern, W.W., The classical ring of quotients of C_{c}(X), Appl. Gen.Topol. 15(2) (2014), 147-154.

[7] Dow, A., Henriksen, M., Kopperman, R., and Vermeer, J., The Space of minimal prime ideals of πΆ(π) need not be basically disconnected, Proc. Amer. Math. Soc. 104(1) (1988), 317-320.

[8] Engelking, R., “General Topology”, Sigma Ser. Pure Math., Heldermann Verlag, Berlin, 1989.

[9] Fine, N., Gilman, L., Extensions of continuous functions in π½N, Bull. Amer. Math. Soc. 66 (1960), 376-381.

[10] Ghadermazi, M., Karamzadeh, O.A.S., and Namdari, M., πΆ(π) versus its functionally countable subalgebra, Bull. Iran Math. Soc. 45(1) (2019), 173-187.

[11] Ghadermazi, M., Karamzadeh, O.A.S., and Namdari, M., On the functionally countable subalgebra of πΆ(π), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-69.

[12] Gillman, L. and Jerison, M., “Rings of Continuous Functions”, Springer-Verlag, Berlin, Heidelberg, New York, 1976.

[13] Henriksen, M. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130.

[14] Henriksen, M. and Woods, R.G., Cozero complemented spaces, when the space of minimal prime ideals of πΆ(π) is compact, Topol. Appl. 141 (2004), 147-170.

[15] Huckaba, J.A. and Keller, J.M., Annihilation of ideals in commutative ring, Pacific J. Math. 83 (1979), 375-379.

[16] Karamzadeh, O.A.S. and Keshtkar, Z., On π-realcompact spaces, Quaest. Math. 41(8) (2018), 1-33.

[17] Keshtkar, Z., Some Disconnected Spaces versus Baer and Rickart on C_{c}(X), Submitted.

[18] Lu, D. and Yu, W., On prime spectrum of a commutative ring, Comm. Algebra 34 (2006), 2667-2672.

[19] Matlis, E., The minimal prime spectrum of a reduced ring, Illinois J. Math. 27 (1983), 353-391.

[20] Namdari, M. and Veisi, A., The subalgebra of Cc(X) consisting of element with countable image versus πΆ(π) with respect to their rings of quotients, Far East J. Math. Sci. 59 (2011), 201-212.

[21] Namdari, M. and Veisi, A., Rings of quotients of the subalgebra of πΆ(π) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571.

[22] Veisi, A., e_{c}-Filters and e_{c} -ideals in the functionally countable subalgebra of C*(X), Appl. Gen. Topol. 20(2) (2019), 395-405.

[23] Veisi, A., On the m_{c}-topology on the functionally countable subalgebra of πΆ(π), J. Algebr. Syst. 9(2) (2022), 335-345.

July 2022

Pages 85-100