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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| Sprache: | Englisch |
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Інститут прикладної математики і механіки НАН України
2010
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862681488098590720 |
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| author | Bhat, V.K. |
| author_facet | Bhat, V.K. |
| 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. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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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| first_indexed | 2025-12-07T15:50:21Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-154506 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:50:21Z |
| publishDate | 2010 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | 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 |
| spellingShingle | Associated prime ideals of weak σ-rigid rings and their extensions Bhat, V.K. |
| title | 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_short | Associated prime ideals of weak σ-rigid rings and their extensions |
| title_sort | associated prime ideals of weak σ-rigid rings and their extensions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154506 |
| work_keys_str_mv | AT bhatvk associatedprimeidealsofweakσrigidringsandtheirextensions |