Associated prime ideals of weak σ-rigid rings and their extensions

• Дата: 2010
• Автор: Bhat, V.K.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут прикладної математики і механіки НАН України 2010
• Назва видання: Algebra and Discrete Mathematics
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/154506
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: 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.

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.