Let m, n be two positive integers, \({\mathbb k}\) be an algebraically closed field with \({\rm char}({\mathbb k}) \, \nmid \, mn\) . Radford constructed an mn2-dimensional Hopf algebra Rmn(q) such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double D(Rmn(q)) of Radford Hopf algebra Rmn(q) has ribbon elements if and only if n is odd. Moreover, if m is even and n is odd, then D(Rmn(q)) has two ribbon elements, if both m and n are odd, then D(Rmn(q)) has only one ribbon element. Moreover, we compute explicitly all ribbon elements of D(Rmn(q)).
