On (σ, δ)(S, 1) rings and their extensions
Let R be a ring, σ an endomorphism of R and δ a σ derivation of R. We recall that R is called an (S, 1)-ring if for a, b _ R, ab = 0 implies aRb = 0. We involve σ and δ to generalize this notion and say that R is a (σ, δ) - (S, 1) ring if for a, b _ R, ab = 0 implies aRb = 0, σ(a)Rb = 0, aR_(b) = 0...
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| Format: | Article |
| Language: | English |
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University of Kragujevac, Faculty of Technical Sciences Čačak, Serbia
2017-01-01
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| Series: | Mathematica Moravica |
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| Online Access: | http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2017/1450-59321701061B.pdf |
| Summary: | Let R be a ring, σ an endomorphism of R and δ a σ derivation of R. We recall that R is called an (S, 1)-ring if for a, b _ R, ab = 0 implies aRb = 0. We involve σ and δ to generalize this notion and say that R is a (σ, δ) - (S, 1) ring if for a, b _ R, ab = 0 implies aRb = 0, σ(a)Rb = 0, aR_(b) = 0 and δ(a)Rb = 0. In case σ is identity, R is called a δ - (S, 1) ring. 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 (Q is the field of rational numbers), σ an automorphism of R and δ a σ-derivation of R such that R is a (σ, δ) - (S, 1) ring. 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 (viewed as a right module over itself) such that (P ∩ R)[x, σ, δ] = P and P ∩ R) = U. |
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| ISSN: | 1450-5932 2560-5542 |