When is the super socle of C(X) prime?
Keywords:super socle of C(X), countably isolated point, countably discrete space, cocountably-disconnected space, one-point Lindelöffication
Let SCF(X) denote the ideal of C(X) consisting of functions which are zero everywhere except on a countable number of points of X. It is generalization of the socle of C(X) denoted by CF(X). Using this concept we extend some of the basic results concerning CF(X) to SCF(X). In particular, we characterize the spaces X such that SCF(X) is a prime ideal in C(X) (note, CF(X) is never a prime ideal in C(X)). This may be considered as an advantage of SCF(X) over C(X). We are also interested in characterizing topological spaces X such that Cc(X) =R+SCF(X), where Cc(X) denotes the subring of C(X) consisting of functions with countable image.
F. Azarpanah, Algebraic properties of some compact spaces, Real Anal. Exchange 25 (2000), 317-328.
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 and O. A. S. Karamzadeh, Algebric characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), 155-168.
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
T. Dube, Contracting the socle in ring 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.
A. A. Estaji and O. A. S. karamzadeh, On C(X) modulo its socle, Comm. Algebra 13 (2003),1561-1571. https://doi.org/10.1081/AGB-120018497
M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, C(X) versus its functionally countable subalgebra, Bull. Iranian Math. Soc. 45 (2019), 173-187. 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
S. Ghasemzadeh, O. A. S. Karamzadeh and M. Namdari, The super socle of the ring of continuous functions, Math. Slovaca 67 (2017), 1001-1010. https://doi.org/10.1515/ms-2017-0028
L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976.
O. A. S. Karamzadeh, M. Motamedi and S. M. Shahrtash, On rings with a unique proper essential right ideal, Fund. Math. 183 (2004), 229-244. https://doi.org/10.4064/fm183-3-3
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/10.1090/s0002-9939-1985-0766552-9
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
S. Mehran and M. Namdari, The λ-super socle of the ring of continuous functions, Categ. General Alg. Struct. Appl. 6 (2017), 37-50.
M. Namdari and M. A. Siavoshi, A note on discrete c-embedded subspaces, Mathematica Slovaca, to appear. https://doi.org/10.1515/ms-2017-0239
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