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Let RR be a right Noetherian ring which is also an algebra over QQ (QQ the field of rational numbers). Let σσ be an automorphism of R and δδ a σσ-derivation of RR. Let further σσ be such that aσ(a)∈P(R)aσ(a)∈P(R) implies that a∈P(R)a∈P(R) for a∈Ra∈R, where P(R)P(R) is the prime radical of RR. In this paper we study minimal prime ideals of Ore extension R[x;σ,δ]R[x;σ,δ] and we prove the following in this direction: Let RR be a right Noetherian ring which is also an algebra over QQ. Let σσ and δδ be as above. Then PP is a minimal prime ideal of R[x;σ,δ]R[x;σ,δ] if and only if there exists a minimal prime ideal UU of RR with P=U[x;σ,δ]P=U[x;σ,δ].