On atomic ideals in some factor rings of $C(X,\Bbb Z)$
Commentationes Mathematicae Universitatis Carolinae, Tome 62 (2021) no. 2, pp. 259-263
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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.
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.
DOI :
10.14712/1213-7243.2021.013
Classification :
54C40
Keywords: $P$-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle
Keywords: $P$-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle
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author = {Olfati, Alireza},
title = {On atomic ideals in some factor rings of $C(X,\Bbb Z)$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {259--263},
year = {2021},
volume = {62},
number = {2},
doi = {10.14712/1213-7243.2021.013},
mrnumber = {4303551},
zbl = {07396222},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2021.013/}
}
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Olfati, Alireza. On atomic ideals in some factor rings of $C(X,\Bbb Z)$. Commentationes Mathematicae Universitatis Carolinae, Tome 62 (2021) no. 2, pp. 259-263. doi: 10.14712/1213-7243.2021.013
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