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

Hani A. Khashan, H. Al-Ezeh (2007)

Archivum Mathematicum

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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 $R\left[x\right]$ is an almost $Q$-ring are equivalent for any ring $R$. Then we prove that under the condition that every prime ideal of $R\left(x\right)$ is an extension of a prime ideal of $R$, the ring $R$...