##
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

** E-mail:**
hakhashan@yahoo.com

**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 $R\langle x\rangle$.