Previous |  Up |  Next


$Q$-rings; almost $Q$-rings; the rings $R(x)$ and $R\langle x\rangle $
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.
[1] Anderson D. D., Mahaney L. A.: Commutative rings in which every ideal is a product of primary ideals. J. Algebra 106 (1987), 528–535. MR 0880975 | Zbl 0607.13004
[2] Anderson D. D., Anderson D. F., Markanda R.: The rings $R(x)$ and $R\left\langle x\right\rangle$. J. Algebra 95 (1985), 96–115. MR 0797658
[3] Heinzer W., David L.: The Laskerian property in commutative rings. J. Algebra 72 (1981), 101–114. MR 0634618 | Zbl 0498.13001
[4] Huckaba J. A.: Commutative rings with zero divisors. Marcel Dekker, INC. New York and Basel, 1988. MR 0938741 | Zbl 0637.13001
[5] Jayaram C.: Almost Q-rings. Arch. Math. (Brno) 40 (2004), 249–257. MR 2107019 | Zbl 1112.13004
[6] Kaplansky I.: Commutative Rings. Allyn and Bacon, Boston 1970. MR 0254021 | Zbl 0203.34601
[7] Larsen M., McCarthy P.: Multiplicative theory of ideals. Academic Press, New York and London 1971. MR 0414528 | Zbl 0237.13002
[8] Ohm J., Pendleton R. L.: Rings with Noetherian spectrum. Duke Math. J. 35 (1968), 631–640. MR 0229627 | Zbl 0172.32202
Partner of
EuDML logo