On the intrinsic topology and some related ideals of 𝐶(𝑋)

Karamzadeh, O. A. S.; Rostami, M.

doi:10.1090/S0002-9939-1985-0766552-9

On the intrinsic topology and some related ideals of $C(X)$
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by O. A. S. Karamzadeh and M. Rostami PDF

Proc. Amer. Math. Soc. 93 (1985), 179-184 Request permission

Abstract:

The above topology is defined and studied on $C(X)$, the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. The minimal ideals and the socle of $C(X)$ are characterized via their corresponding $z$-filters. We observe that these ideals are $z$-ideals and $X$ is discrete if and only if the socle of $C(X)$ is a free ideal. It is also shown that for a class of topological spaces, containing all $P$-spaces, the family ${C_k}(X)$ of functions with compact support is identical with the intersection of the free maximal ideals of $C(X)$.

References

Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
J. Glenn Brookshear, On projective prime ideals in $C(X)$, Proc. Amer. Math. Soc. 69 (1978), no. 1, 203–204. MR 470929, DOI 10.1090/S0002-9939-1978-0470929-5
W. W. Comfort, On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107–118. MR 222846, DOI 10.1090/S0002-9947-1968-0222846-1
Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
Leonard Gillman, Countably generated ideals in rings of continuous functions, Proc. Amer. Math. Soc. 11 (1960), 660–666. MR 156189, DOI 10.1090/S0002-9939-1960-0156189-X
Oscar Goldman, A Wedderburn-Artin-Jacobson structure theorem, J. Algebra 34 (1975), 64–73. MR 366975, DOI 10.1016/0021-8693(75)90193-3
R. C. Haworth and R. A. McCoy, Baire spaces, Dissertationes Math. (Rozprawy Mat.) 141 (1977), 73. MR 431104
M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130. MR 194880, DOI 10.1090/S0002-9947-1965-0194880-9
Irving Kaplansky, Topological rings, Amer. J. Math. 69 (1947), 153–183. MR 19596, DOI 10.2307/2371662
O. A. S. Karamzadeh, Projective maximal right ideals of self-injective rings, Proc. Amer. Math. Soc. 48 (1975), 286–288. MR 360705, DOI 10.1090/S0002-9939-1975-0360705-3
O. A. S. Karamzadeh and M. Motamedi, A note on rings in which every maximal ideal is generated by a central idempotent, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 3, 124–125. MR 664552
O. A. S. Karamzadeh, On maximal right ideals which are direct summands, Bull. Iranian Math. Soc. 10 (1983), no. 1-2, 40–46. MR 738689
O. A. S. Karamzadeh, On maximal right ideals which are direct summands, Bull. Iranian Math. Soc. 10 (1983), no. 1-2, 40–46. MR 738689
O. A. S. Karamzadeh, On the classical Krull dimension of rings, Fund. Math. 117 (1983), no. 2, 103–108. MR 719833, DOI 10.4064/fm-117-2-103-108
C. W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45 (1957), 28–50. MR 102731, DOI 10.4064/fm-45-1-28-50

Rings and modules

Mark Mandelker, Supports of continuous functions, Trans. Amer. Math. Soc. 156 (1971), 73–83. MR 275367, DOI 10.1090/S0002-9947-1971-0275367-4
G. De Marco, On the countably generated $z$-ideals of $C(X)$, Proc. Amer. Math. Soc. 31 (1972), 574–576. MR 288563, DOI 10.1090/S0002-9939-1972-0288563-3
M. Rajagopalan and R. F. Wheeler, Sequential compactness of $X$ implies a completeness property for $C(X)$, Canadian J. Math. 28 (1976), no. 1, 207–210. MR 405341, DOI 10.4153/CJM-1976-026-9
Stewart M. Robinson, The intersection of the free maximal ideals in a complete space, Proc. Amer. Math. Soc. 17 (1966), 468–469. MR 188974, DOI 10.1090/S0002-9939-1966-0188974-8
David Rudd, $P$-ideals and $F$-ideals in rings of continuous functions, Fund. Math. 88 (1975), no. 1, 53–59. MR 380716, DOI 10.4064/fm-88-1-53-59
Walter Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39–42. MR 85475, DOI 10.1090/S0002-9939-1957-0085475-7
Robert F. Wheeler, The strict topology for $P$-spaces, Proc. Amer. Math. Soc. 41 (1973), 466–472. MR 341048, DOI 10.1090/S0002-9939-1973-0341048-9
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581