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

H. A. Khashan and H. Al-Ezeh

Address. H. A. Khashan, Department of Mathematics, Al al-Bayt University, Al-Mafraq 130095, Jordan
H. Al-Ezeh, Department of Mathematics, University of Jordan, Amman 11942, Jordan


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.

AMSclassification. 13A15.

Keywords. $Q$-rings, almost $Q$-rings, the rings $R(x)$ and $R\langle x\rangle$.