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 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 .