On the essentiality and primeness of λ-super socle of C(X)

Authors

  • S. Mehran Islamic Azad University
  • M. Namdari Shahid Chamran University of Ahvaz
  • S. Soltanpour Petroleum University of Technology

DOI:

https://doi.org/10.4995/agt.2018.9058

Keywords:

λ-super socle of C(X), λ-isolated point, λ-disjoint spaces

Abstract

Spaces X for which the annihilator of Sλ(X), the λ-super socle of C(X) (i.e., the set of elements of C(X) that cardinality of their cozerosets are less than λ, where λ is a regular cardinal number such that λ≤|X|) is generated by an idempotent are characterized. This enables us to find a topological property equivalent to essentiality of Sλ(X). It is proved that every prime ideal in C(X) containing Sλ(X) is essential and it is an intersection of free prime ideals. Primeness of Sλ(X) is characterized via a fixed maximal ideal of C(X).

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Author Biographies

S. Mehran, Islamic Azad University

Shoushtar Branch

Department of Mathematics

M. Namdari, Shahid Chamran University of Ahvaz

Department of Mathematics

References

F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar. 31 (1995), 105-112. https://doi.org/10.1007/BF01876485

F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 2149-2154. https://doi.org/10.1090/S0002-9939-97-04086-0

F. Azarpanah, Algebraic properties of some compact spaces, Real Analysis Exchange 25, no. 1 (1999), 317-328.

F. Azarpanah, O. A. S. Karamzadeh and S. Rahmati, C(X) VS. C(X) modulo its socle, Coll. Math. 3 (2008), 315-336. https://doi.org/10.4064/cm111-2-9

F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar A. R. Olfati, On maximal ideals of Cc(X) and the uniformity of its localizations, Rocky Mountain Journal of Mathematics 48, no. 2 (2018), 345-384. https://doi.org/10.1216/RMJ-2018-48-2-345

T. Dube, Contracting the socle in rings of continuous functions, Rend. Semin. Mat. Univ. Padova {123} (2010), 37-53. https://doi.org/10.4171/RSMUP/123-2

R. Engelking, General topology, Heldermann Verlag Berlin, 1989.

S. G. Ghasemzadeh, O. A. S. Karamzadeh and M. Namdari, The super socle of the ring of continuous functions, Math. Slovaca 67, no. 4 (2017), 1-10. https://doi.org/10.1515/ms-2017-0028

M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, C(X) versus its functionally countable subalgebra, Bull. Iranian Math. Soc. https://doi.org/10.1007/s41980-018-0124-8

M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-70. https://doi.org/10.4171/RSMUP/129-4

L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976.

K. R. Goodearl, JR. and R. B. Warfiel, An introduction to noncommutative Noetherianrings, London Mathematical Society Students Texts, Vol. 16, Cambridge UniversityPress, 1989.

K. R. Goodearl, Von Neumann regular rings, Pitman Publishing Limited, London, SanFrancisco, Melbourne, 1979.

O. A. S. Karamzadeh, M. Namdari and M. A. Siavoshi, A note on $lambda$-compact spaces, Math. Slovaca 63, no. 6 (2013), 1371-1380. https://doi.org/10.2478/s12175-013-0177-3

O. A. S. Karamzadeh, M. Namdari and S. Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16, no. 2 (2015), 183-207. https://doi.org/10.4995/agt.2015.3445

O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184. https://doi.org/10.2307/2044578

S. Mehran and M. Namdari, The λ- super socle of the ring of continuous functions, Categories and General Algebraic Structures with Applications 6, no. 1 (2017), 37-50.

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Published

2018-10-04

How to Cite

[1]
S. Mehran, M. Namdari, and S. Soltanpour, “On the essentiality and primeness of λ-super socle of C(X)”, Appl. Gen. Topol., vol. 19, no. 2, pp. 261–268, Oct. 2018.

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