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Conditions under which \(R(x)\) and \(R\langle x\rangle \) are almost \(Q\)-rings. (English) Zbl 1155.13301

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
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