Document Type: Research Paper

**Author**

Departemant of Mathematics, Shahid Chamran University, Ahvaz, Iran

**Abstract**

In this paper, we introduce and study a mapping from the collection of all intermediate rings of $C(X)$ to the collection of all realcompactifications of $X$ contained in $\beta X$. By establishing the relations between this mapping and its converse, we give a different approach to the main statements of De et. al.

Using these, we provide different answers to the four basic questions raised in Acharyya et.al. Finally, we give some notes on the realcompactifications generated by ideals.

Using these, we provide different answers to the four basic questions raised in Acharyya et.al. Finally, we give some notes on the realcompactifications generated by ideals.

**Keywords**

[2] Acharyya, S.K., Chattopadhyay, K.C., and Ghosh, D.P., On a class of subagebrasof C(X) and the intersection of their free maximal ideals, Proc. Amer. Math. Soc.125 (1997), 611-615.

[3] Acharyya, S.K., and De, D., An interesting class of ideals in subalgebras of C(X)containing C*(X), Comment. Math. Univ. Carolin. 48 (2007), 273-280.

[4] Acharyya, S.K. and De, D., A-compactifications and minimal subalgebras of C(X),Rocky Mountain J. Math. 35 (2005), 1061-1067.

[5] Aliabad, A.R. and Parsinia, M., zR-ideals and z_R-ideals in subrings of RX, IranianJ. Math. Sci. Inform., to appaer.

[6] Aliabad, A.R. and Parsinia, M., Remarks on subrings of C(X) of the form I+C*(X),Quaest. Math. 40(1) (2017), 63-73.

[7] Azarpanah, F. and Mohamadian, R.,pz-ideals andpz_-ideals in C(X), Acta. Math.Sin. (Eng. Ser.) 23 (2007), 989-006.

[8] De, D. and Acharyya, S.K., Characterization of function rings between C*(X) andC(X), Kyungpook Math. J. 46 (2006), 503-507.

[9] Dominguez, J.M. and Gomez-Perez, J., There do not exist minimal algebras betweenC*(X) and C(X) with prescribed real maximal ideal space, Acta. Math. Hungar.94 (2002), 351-355.

[10] Dominguez, J.M., Gomez, J., and Mulero, M.A., Intermediate algebras betweenC*(X) and C(X) as ring of fractions of C*(X), Topology Appl. 77 (1997), 115-130.

[11] Gillman, L. and Jerison, M., “Rings of Continuous Functions”, Springer-Verlag, 1978.

[12] Parsinia, M., Remarks on LBI-subalgebras of C(X), Comment. Math. Univ. Carolin.57 (2016), 261-270.

[13] Parsinia, M., Remarks on intermediate C-rings of C(X), Quaest. Math., Publishedonline (2017).

[14] Parsinia, M., R-P-spaces and subrings of C(X), Filomat 32(1) (2018), 319–328.

[15] Plank, D., On a class of subalgebras of C(X) with applications to FX X, Fund.Math., 64 (1969), 41-54.

[16] Redlin, L. and Watson, S., Structure spaces for rings of continuous functions withapplications to realcompactifications, Fund. Math. 152 (1997), 151-163.

[17] Rudd, D., On two sum theorems for ideals in C(X), Michigan Math. J. 19 (1970),139-141.

[18] Sack, J. and Watson, S., C and C* among intermediate rings, Topology Proc. 43(2014), 69-82.

Volume 10, Issue 1

Winter and Spring 2019

Pages 107-116