Conditions under which $R\left(x\right)$ and $R\langle x\rangle$ are almost Q-rings

Khashan, Hani A., and Al-Ezeh, H.. "Conditions under which $R(x)$ and $R\langle x\rangle $ are almost Q-rings." Archivum Mathematicum 043.4 (2007): 231-236. <http://eudml.org/doc/250149>.

@article{Khashan2007,
abstract = {All rings considered in this paper are assumed to be commutative with identities. A ring $R$ is a $Q$-ring if every ideal of $R$ is a finite product of primary ideals. An almost $Q$-ring is a ring whose localization at every prime ideal is a $Q$-ring. In this paper, we first prove that the statements, $R$ is an almost $ZPI$-ring and $R[x]$ is an almost $Q$-ring are equivalent for any ring $R$. Then we prove that under the condition that every prime ideal of $R(x)$ is an extension of a prime ideal of $R$, the ring $R$ is a (an almost) $Q$-ring if and only if $R(x)$ is so. Finally, we justify a condition under which $R(x)$ is an almost $Q$-ring if and only if $R\left\langle x\right\rangle $ is an almost $Q$-ring.},
author = {Khashan, Hani A., Al-Ezeh, H.},
journal = {Archivum Mathematicum},
keywords = {$Q$-rings; almost $Q$-rings; the rings $R(x)$ and $R\langle x\rangle $; -rings},
language = {eng},
number = {4},
pages = {231-236},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Conditions under which $R(x)$ and $R\langle x\rangle $ are almost Q-rings},
url = {http://eudml.org/doc/250149},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Khashan, Hani A.
AU - Al-Ezeh, H.
TI - Conditions under which $R(x)$ and $R\langle x\rangle $ are almost Q-rings
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 4
SP - 231
EP - 236
AB - All rings considered in this paper are assumed to be commutative with identities. A ring $R$ is a $Q$-ring if every ideal of $R$ is a finite product of primary ideals. An almost $Q$-ring is a ring whose localization at every prime ideal is a $Q$-ring. In this paper, we first prove that the statements, $R$ is an almost $ZPI$-ring and $R[x]$ is an almost $Q$-ring are equivalent for any ring $R$. Then we prove that under the condition that every prime ideal of $R(x)$ is an extension of a prime ideal of $R$, the ring $R$ is a (an almost) $Q$-ring if and only if $R(x)$ is so. Finally, we justify a condition under which $R(x)$ is an almost $Q$-ring if and only if $R\left\langle x\right\rangle $ is an almost $Q$-ring.
LA - eng
KW - $Q$-rings; almost $Q$-rings; the rings $R(x)$ and $R\langle x\rangle $; -rings
UR - http://eudml.org/doc/250149
ER -