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.
@article{108067, author = {Hani A. Khashan and H. Al-Ezeh}, title = {Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings}, journal = {Archivum Mathematicum}, volume = {043}, year = {2007}, pages = {231-236}, zbl = {1155.13301}, mrnumber = {2378523}, language = {en}, url = {http://dml.mathdoc.fr/item/108067} }
Khashan, Hani A.; Al-Ezeh, H. Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings. Archivum Mathematicum, Tome 043 (2007) pp. 231-236. http://gdmltest.u-ga.fr/item/108067/
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