Geodesic
Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1049-1056
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma $ be an automorphism of $R$ and $\delta $ a $\sigma $-derivation of $R$. Then $R$ is said to be an almost $\delta $-divided ring if every minimal prime ideal of $R$ is $\delta $-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb {Q}$ ($\mathbb {Q}$ is the field of rational numbers). Let $\sigma $ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta $ a $\sigma $-derivation of $R$ such that $\sigma (\delta (a)) = \delta (\sigma (a))$ for all $a \in R$. Further, if for any strongly prime ideal $U$ of $R$ with $\sigma (U) = U$ and $\delta (U)\subseteq \delta $, $U[x; \sigma , \delta ]$ is a strongly prime ideal of $R[x; \sigma , \delta ]$, then we prove the following: (1) $R$ is a near pseudo valuation ring if and only if the Ore extension $R[x; \sigma ,\delta ]$ is a near pseudo valuation ring. (2) $R$ is an almost $\delta $-divided ring if and only if $R[x;\sigma ,\delta ]$ is an almost $\delta $-divided ring.
Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma $ be an automorphism of $R$ and $\delta $ a $\sigma $-derivation of $R$. Then $R$ is said to be an almost $\delta $-divided ring if every minimal prime ideal of $R$ is $\delta $-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb {Q}$ ($\mathbb {Q}$ is the field of rational numbers). Let $\sigma $ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta $ a $\sigma $-derivation of $R$ such that $\sigma (\delta (a)) = \delta (\sigma (a))$ for all $a \in R$. Further, if for any strongly prime ideal $U$ of $R$ with $\sigma (U) = U$ and $\delta (U)\subseteq \delta $, $U[x; \sigma , \delta ]$ is a strongly prime ideal of $R[x; \sigma , \delta ]$, then we prove the following: (1) $R$ is a near pseudo valuation ring if and only if the Ore extension $R[x; \sigma ,\delta ]$ is a near pseudo valuation ring. (2) $R$ is an almost $\delta $-divided ring if and only if $R[x;\sigma ,\delta ]$ is an almost $\delta $-divided ring.
DOI : 10.1007/s10587-013-0071-8
Classification : 16N40, 16P40, 16S36
Keywords: Ore extension; automorphism; derivation; minimal prime; pseudo-valuation ring; near pseudo-valuation ring 
@article{10_1007_s10587_013_0071_8,
     author = {Bhat, Vijay Kumar},
     title = {Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1049--1056},
     year = {2013},
     volume = {63},
     number = {4},
     doi = {10.1007/s10587-013-0071-8},
     mrnumber = {3165514},
     zbl = {1299.16020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0071-8/}
}
TY  - JOUR
AU  - Bhat, Vijay Kumar
TI  - Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings
JO  - Czechoslovak Mathematical Journal
PY  - 2013
SP  - 1049
EP  - 1056
VL  - 63
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0071-8/
DO  - 10.1007/s10587-013-0071-8
LA  - en
ID  - 10_1007_s10587_013_0071_8
ER  - 
%0 Journal Article
%A Bhat, Vijay Kumar
%T Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings
%J Czechoslovak Mathematical Journal
%D 2013
%P 1049-1056
%V 63
%N 4
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0071-8/
%R 10.1007/s10587-013-0071-8
%G en
%F 10_1007_s10587_013_0071_8
Bhat, Vijay Kumar. Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1049-1056. doi: 10.1007/s10587-013-0071-8

[1] Anderson, D. F.: Comparability of ideals and valuation overrings. Houston J. Math. 5 (1979), 451-463. | MR | Zbl

[2] Anderson, D. F.: When the dual of an ideal is a ring. Houston J. Math. 9 (1983), 325-332. | MR | Zbl

[3] Badawi, A.: On divided commutative rings. Commun. Algebra 27 (1999), 1465-1474. | DOI | MR | Zbl

[4] Badawi, A.: On domains which have prime ideals that are linearly ordered. Commun. Algebra 23 (1995), 4365-4373. | DOI | MR | Zbl

[5] Badawi, A.: On $\phi$-pseudo-valuation Rings. Advances in Commutative Ring Theory. D. E. Dobbs et al. Proceedings of the 3rd International Conference, Fez, Morocco Lect. Notes Pure Appl. Math. 205 Marcel Dekker, New York (1999), 101-110. | MR

[6] Badawi, A.: On pseudo-almost valuation domains. Commun. Algebra 35 (2007), 1167-1181. | DOI | MR | Zbl

[7] Badawi, A., Anderson, D. F., Dobbs, D. E.: Pseudo-valuation Rings. Commutative Ring Theory. P.-J. Cahen et al. Proceedings of the 2nd International Conference, Fes, Morocco, June 5-10, 1995. Lect. Notes Pure Appl. Math. 185 Marcel Dekker, New York (1997), 57-67. | MR

[8] Badawi, A., Houston, E.: Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conducive domains. Commun. Algebra 30 (2002), 1591-1606. | DOI | MR | Zbl

[9] Bell, H. E., Mason, G.: On derivations in near-rings and rings. Math. J. Okayama Univ. 34 (1992), 135-144. | MR | Zbl

[10] Bhat, V. K.: A note on completely prime ideals of Ore extensions. Int. J. Algebra Comput. 20 (2010), 457-463. | DOI | MR | Zbl

[11] Bhat, V. K.: On near pseudo valuation rings and their extensions. Int. Electron. J. Algebra (electronic only) 5 (2009), 70-77. | MR | Zbl

[12] Bhat, V. K.: Polynomial rings over pseudovaluation rings. Int. J. Math. Math. Sci. 2007 (2007), Article ID 20138, 6 pages. | MR | Zbl

[13] Bhat, V. K., Kumari, N.: On Ore extensions over near pseudo valuation rings. Int. J. Math. Game Theory Algebra 20 (2011), 69-77. | MR

[14] Bhat, V. K., Kumari, N.: Transparency of $\sigma(\ast)$-rings and their extensions. Int. J. Algebra 2 (2008), 919-924. | MR

[15] Goodearl, K. R., Jr., R. B. Warfield: An Introduction to Noncommutative Noetherian Rings. London Mathematical Society Student Texts 16 Cambridge University Press, Cambridge (1989). | MR | Zbl

[16] Hedstrom, J. R., Houston, E. G.: Pseudo-valuation domains. Pac. J. Math. 75 (1978), 137-147. | DOI | MR | Zbl

[17] Krempa, J.: Some examples of reduced rings. Algebra Colloq. 3 (1996), 289-300. | MR | Zbl

[18] Kwak, T. K.: Prime radicals of skew polynomial rings. Int. J. Math. Sci. 2 (2003), 219-227. | MR | Zbl

[19] McConnell, J. C., Robson, J. C.: Noncommutative Noetherian Rings. With the cooperation of L. W. Small. Reprinted with corrections from the 1987 original. Graduate Studies in Mathematics 30 American Mathematical Society, Providence (2001). | MR | Zbl

Cité par Sources :