Associated prime ideals of weak σ-rigid rings and their extensions
Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements of R. In this paper we...
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| Veröffentlicht in: | Algebra and Discrete Mathematics |
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| Datum: | 2010 |
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Інститут прикладної математики і механіки НАН України
2010
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| Zitieren: | Associated prime ideals of weak σ-rigid rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 8–17. — Бібліогр.: 15 назв. — англ. |
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Bhat, V.K. 2019-06-15T16:14:20Z 2019-06-15T16:14:20Z 2010 Associated prime ideals of weak σ-rigid rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 8–17. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16-XX; 16N40, 16P40, 16S36. https://nasplib.isofts.kiev.ua/handle/123456789/154506 Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements of R. In this paper we study the associated prime ideals of Ore extension R[x;σ,δ] and we prove the following in this direction: Let R be a semiprime right Noetherian ring which is also an algebra over Q. Let σ and δ be as above. Then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R with σ(U)=U and δ(U)⊆U and P=U[x;σ,δ]. We also prove that if R be a right Noetherian ring which is also an algebra over Q, σ and δ as usual such that σ(δ(a))=δ(σ(a)) for all a∈R and σ(U)=U for all associated prime ideals U of R (viewed as a right module over itself), then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R such that (P∩R)[x;σ,δ]=P and P∩R=U. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Associated prime ideals of weak σ-rigid rings and their extensions Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Associated prime ideals of weak σ-rigid rings and their extensions |
| spellingShingle |
Associated prime ideals of weak σ-rigid rings and their extensions Bhat, V.K. |
| title_short |
Associated prime ideals of weak σ-rigid rings and their extensions |
| title_full |
Associated prime ideals of weak σ-rigid rings and their extensions |
| title_fullStr |
Associated prime ideals of weak σ-rigid rings and their extensions |
| title_full_unstemmed |
Associated prime ideals of weak σ-rigid rings and their extensions |
| title_sort |
associated prime ideals of weak σ-rigid rings and their extensions |
| author |
Bhat, V.K. |
| author_facet |
Bhat, V.K. |
| publishDate |
2010 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements of R. In this paper we study the associated prime ideals of Ore extension R[x;σ,δ] and we prove the following in this direction:
Let R be a semiprime right Noetherian ring which is also an algebra over Q. Let σ and δ be as above. Then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R with σ(U)=U and δ(U)⊆U and P=U[x;σ,δ].
We also prove that if R be a right Noetherian ring which is also an algebra over Q, σ and δ as usual such that σ(δ(a))=δ(σ(a)) for all a∈R and σ(U)=U for all associated prime ideals U of R (viewed as a right module over itself), then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R such that (P∩R)[x;σ,δ]=P and P∩R=U.
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| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/154506 |
| citation_txt |
Associated prime ideals of weak σ-rigid rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 8–17. — Бібліогр.: 15 назв. — англ. |
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2025-12-07T15:50:21Z |
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2025-12-07T15:50:21Z |
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1850865218252439552 |