Khashan, H. A.; Al-Ezeh, H. Conditions under which \(R(x)\) and \(R\langle x\rangle \) are almost \(Q\)-rings. (English) Zbl 1155.13301 Arch. Math., Brno 43, No. 4, 231-236 (2007). Summary: 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. MSC: 13A15 Ideals and multiplicative ideal theory in commutative rings Keywords:\(Q\)-rings PDFBibTeX XMLCite \textit{H. A. Khashan} and \textit{H. Al-Ezeh}, Arch. Math., Brno 43, No. 4, 231--236 (2007; Zbl 1155.13301) Full Text: EuDML EMIS