2-primal Ore extensions over weak σ-rigid rings

Gosani, Smarti and Bhat, V. K. (2015) 2-primal Ore extensions over weak σ-rigid rings. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 31 (2). pp. 227-232. ISSN 0866-0174

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Abstract

Let R be a ring, σ an automorphism of R and δ a σ-derivation of R. We recall that a ring R is said to be a δ-ring if aδ(a) ∈ P(R) implies a ∈ P(R), where P(R) denotes the prime radical of R. It is known that, if R is a Noetherian ring, σ an automorphism of R such that aσ(a) ∈ P(R) implies a ∈ P(R) and δ a σ-derivation of R such that R is a δ-ring with σ(δ(a)) = δ(σ(a)), for all a ∈ R, then R[x; σ, δ] is a 2- primal Noetherian ring. We investigate this result if P(R) is replaced with N(R) and prove that if R is a Noetherian ring, which is also an algebra over Q, σ an automorphism of R such that aσ(a) ∈ N(R) if and only if a ∈ N(R), where N(R) denotes the set of nilpotent elements of R and δ a σ-derivation of R such that R is a δ-ring with δ(P(R)) ⊆ P(R), then R[x; σ, δ] is 2-primal.

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