In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R = ℤ 4 + uℤ 4 ; u 2 = 1, with an automorphism θ and a derivation δ θ . We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as δ θ -cyclic codes. Some properties of skew polynomial ring R[x, θ, δ θ ] are presented. A δ θ -cyclic code is proved to be a left R[x, θ, δ θ ]-submodule of R[x,θ,δ 〈xn −1 〉 θ] . The form of a parity-check matrix of a free δ θ -cyclic codes of even length n is presented. These codes are further generalized to double δ θ -cyclic codes over R. We have obtained some new good codes over ℤ 4 via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of ℤ 4 -codes . © 2018 AIMS.