On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 4، دوره 17، شماره 1، مهر 2022، صفحه 85-100 اصل مقاله (386.14 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.52547/cgasa.2022.102622 | ||
| نویسندگان | ||
| Zahra Keshtkar؛ Rostam Mohamadian* ؛ Mehrdad Namdari؛ Maryam Zeinali | ||
| Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran. | ||
| چکیده | ||
| In this article we consider some relations between the topological properties of the spaces X and Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of Min(Cc (X)) is equivalent to the von-Neumann regularity of qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of Cc(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if 𝑋 is an Fc-space, then Min(Cc (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with β0X. Finally, by introducing zoc-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected. | ||
| کلیدواژهها | ||
| The space of minimal prime ideals؛ strongly zero-dimensional space؛ countably basically disconnected space؛ countably cozero complemented space؛ $z^0_c$-ideals | ||
| مراجع | ||
|
[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 z0-ideals in 𝐶(𝑋), 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 Cc(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 Cc(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 𝐶(𝑋) 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 𝛽N, Bull. Amer. Math. Soc. 66 (1960), 376-381.
[10] Ghadermazi, M., Karamzadeh, O.A.S., and Namdari, M., 𝐶(𝑋) 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 𝐶(𝑋), 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 𝐶(𝑋) 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 𝑐-realcompact spaces, Quaest. Math. 41(8) (2018), 1-33.
[17] Keshtkar, Z., Some Disconnected Spaces versus Baer and Rickart on Cc(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 𝐶(𝑋) 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 𝐶(𝑋) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571.
[22] Veisi, A., ec-Filters and ec -ideals in the functionally countable subalgebra of C*(X), Appl. Gen. Topol. 20(2) (2019), 395-405.
[23] Veisi, A., On the mc-topology on the functionally countable subalgebra of 𝐶(𝑋), J. Algebr. Syst. 9(2) (2022), 335-345. | ||
|
آمار تعداد مشاهده مقاله: 539 تعداد دریافت فایل اصل مقاله: 935 |
||