Geodesic
Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings
Archivum mathematicum, Tome 43 (2007) no. 4, pp. 231-236 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 13A15
Keywords: $Q$-rings; almost $Q$-rings; the rings $R(x)$ and $R\langle x\rangle $ 
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     author = {Khashan, H. A. and Al-Ezeh, H.},
     title = {Conditions under which $R(x)$ and $R\langle x\rangle$ are almost {Q-rings}},
     journal = {Archivum mathematicum},
     pages = {231--236},
     year = {2007},
     volume = {43},
     number = {4},
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     zbl = {1155.13301},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2007_43_4_a0/}
}
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Khashan, H. A.; Al-Ezeh, H. Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings. Archivum mathematicum, Tome 43 (2007) no. 4, pp. 231-236. http://geodesic.mathdoc.fr/item/ARM_2007_43_4_a0/

[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 | Zbl

[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

[3] Heinzer W., David L.: The Laskerian property in commutative rings. J. Algebra 72 (1981), 101–114. | MR | Zbl

[4] Huckaba J. A.: Commutative rings with zero divisors. Marcel Dekker, INC. New York and Basel, 1988. | MR | Zbl

[5] Jayaram C.: Almost Q-rings. Arch. Math. (Brno) 40 (2004), 249–257. | MR | Zbl

[6] Kaplansky I.: Commutative Rings. Allyn and Bacon, Boston 1970. | MR | Zbl

[7] Larsen M., McCarthy P.: Multiplicative theory of ideals. Academic Press, New York and London 1971. | MR | Zbl

[8] Ohm J., Pendleton R. L.: Rings with Noetherian spectrum. Duke Math. J. 35 (1968), 631–640. | MR | Zbl