Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
$P$-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle
$P$-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle
Summary:
A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a, b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ and $C_c(X)/C_F(X)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring $\Bbb{Z}^X/ \Bbb{Z}^{(X)}$ has no atomic ideal. Another result is that for each infinite $P$-space $X$, the socle of the factor ring $C_c(X)/C_F(X)$ is always equal to zero. Also, zero-dimensional spaces $X$ are characterized for which $C^F(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ have atomic ideals.
References:
[1] Alling N. L.: Rings of continuous integer-valued functions and nonstandard arithmetic. Trans. Amer. Math. Soc. 118 (1965), 498–525. DOI 10.1090/S0002-9947-1965-0184960-6 | MR 0184960
[2] Azarpanah F., Karamzadeh O. A. S., Keshtkar Z., Olfati A. R.: On maximal ideals of $C_c(X)$ and the uniformity of its localizations. Rocky Mountain J. Math. 48 (2018), no. 2, 345–384. DOI 10.1216/RMJ-2018-48-2-345 | MR 3809150
[3] Azarpanah F., Karamzadeh O. A. S., Rahmati S.: $C(X)$ vs. $C(X)$ modulo its socle. Colloq. Math. 111 (2008), no. 2, 315–336. DOI 10.4064/cm111-2-9 | MR 2365803
[4] Gillman L., Jerison M.: Rings of Continuous Functions. Graduate Texts in Mathematics, 43, Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[5] Karamzadeh O. A. S., Rostami M.: On the intrinsic topology and some related ideals of $C(X)$. Proc. Amer. Math. Soc. 93 (1985), no. 1, 179–184. MR 0766552 | Zbl 0524.54013
[6] Martinez J.: $C(X, \mathbb{Z})$ revisited. Adv. Math. 99 (1993), no. 2, 152–161. DOI 10.1006/aima.1993.1022 | MR 1219582
[7] Momtahan E., Motamedi M.: A study on dimensions of modules. Bull. Iranian. Math. Soc. 43 (2017), no. 5, 1227–1235. MR 3730636
[8] Mozaffarikhah A., Momtahan E., Olfati A. R., Safaeeyan S.: $p$-semisimple modules and type submodules. J. Algebra Appl. 19 (2020), no. 4, 2050078, 22 pages. DOI 10.1142/S0219498820500784 | MR 4098942
[9] Olfati A. R.: Homomorphisms from $C(X, \mathbb{Z})$ into a ring of continuous functions. Algebra Universalis 79 (2018), no. 2, Paper No. 34, 26 pages. DOI 10.1007/s00012-018-0509-9 | MR 3788813
[10] Pierce R. S.: Rings of integer-valued continuous functions. Trans. Amer. Math. Soc. 100 (1961), 371–394. DOI 10.1090/S0002-9947-1961-0131438-8 | MR 0131438 | Zbl 0196.15401
Search
Partner of
