On atomic ideals in some factor rings of $C(X,\mathbb{Z})$

@article{Olfati2021,
abstract = {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 $C(X,\mathbb \{Z\})/C_F(X,\mathbb \{Z\})$ 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 $\mathbb \{Z\}^X/ \mathbb \{Z\}^\{(X)\}$ 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 $C^F(X,\mathbb \{Z\})/C_F(X,\mathbb \{Z\})$ have atomic ideals.},
author = {Olfati, Alireza},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$P$-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle},
language = {eng},
number = {2},
pages = {259-263},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On atomic ideals in some factor rings of $C(X,\mathbb \{Z\})$},
url = {http://eudml.org/doc/297939},
year = {2021},
}

TY - JOUR
AU - Olfati, Alireza
TI - On atomic ideals in some factor rings of $C(X,\mathbb {Z})$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 2
SP - 259
EP - 263
AB - 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 $C(X,\mathbb {Z})/C_F(X,\mathbb {Z})$ 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 $\mathbb {Z}^X/ \mathbb {Z}^{(X)}$ 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 $C^F(X,\mathbb {Z})/C_F(X,\mathbb {Z})$ have atomic ideals.
LA - eng
KW - $P$-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle
UR - http://eudml.org/doc/297939
ER -