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Vietnam Journal of Mathematics 36:4(2008) 455-461

 Ore Extensions over 2-primal Rings

V. K. Bhat and Ravi Raina

Abstract.  Let R be a ring, σ an automorphism of R and let δ be a σ-derivation of R. 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 δ-Noetherian Q-algebra, σ and δ are as usual such that σ(δ(a)) = δ(σ(a)), for all a  R and σ(P) = P, for all minimal prime ideals P of R, then R[x, σ, δ] is a 2-primal Noetherian ring. In this article it is proved that in the case δ  is the zero map, R is a 2-primal Noetherian ring implies that R[x, σ] is a 2-primal Noetherian ring. In the case σ is the identity map, a similar result is proved for the differential operator ring R[x, δ] (R in this case is moreover a Noetherian Q-algebra).

1991 Mathematics Subject Classification: Primary 16XX, Secondary 16N40, 16P40, 16W20, 16W25.

Keywords: 2-primal, Minimal prime, prime radical, nil radical, automorphism, derivation.

 

 

 

 

 

 

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