Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings
Khashan, Hani A. ; Al-Ezeh, H.
Archivum Mathematicum, Tome 043 (2007), p. 231-236 / Harvested from 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.

Publié le : 2007-01-01
Classification:  13A15 
@article{108067,
     author = {Hani A. Khashan and H. Al-Ezeh},
     title = {Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings},
     journal = {Archivum Mathematicum},
     volume = {043},
     year = {2007},
     pages = {231-236},
     zbl = {1155.13301},
     mrnumber = {2378523},
     language = {en},
     url = {http://dml.mathdoc.fr/item/108067}
}

Khashan, Hani A.; Al-Ezeh, H. Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings. Archivum Mathematicum, Tome 043 (2007) pp. 231-236. http://gdmltest.u-ga.fr/item/108067/

Anderson D. D.; Mahaney L. A. Commutative rings in which every ideal is a product of primary ideals, J. Algebra 106 (1987), 528–535. (1987) | MR 0880975 | Zbl 0607.13004

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. (1985) | MR 0797658

Heinzer W.; David L. The Laskerian property in commutative rings, J. Algebra 72 (1981), 101–114. (1981) | MR 0634618 | Zbl 0498.13001

Huckaba J. A. Commutative rings with zero divisors, Marcel Dekker, INC. New York and Basel, 1988. (1988) | MR 0938741 | Zbl 0637.13001

Jayaram C. Almost Q-rings, Arch. Math. (Brno) 40 (2004), 249–257. | MR 2107019 | Zbl 1112.13004

Kaplansky I. Commutative Rings, Allyn and Bacon, Boston 1970. (1970) | MR 0254021 | Zbl 0203.34601

Larsen M.; Mccarthy P. Multiplicative theory of ideals, Academic Press, New York and London 1971. (1971) | MR 0414528 | Zbl 0237.13002

Ohm J.; Pendleton R. L. Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631–640. (1968) | MR 0229627 | Zbl 0172.32202