Constacyclic codes of length (pr,ps) over mixed alphabets

For any prime p and positive integer m , let R be the finite commutative ring F p m + u F p m + v F p m + u v F p m , where u 2 = 0 , v 2 = 0 and u v = v u . Let λ = λ 1 + u λ 2 + v λ 3 + u v λ 4 be a unit of R , where λ 1 , λ 2 , λ 3 , λ 4 ∈ F p m and λ 1 ≠ 0 . We know that λ -constacyclic codes of length p s over R are exactly ideals of the ring R [ x ] ⟨ x p s - λ ⟩ . For all possible values of λ , we study λ -constacyclic codes of length p s over R . We also extend structures of codes from single alphabet to mixed alphabet, and determine separable constacyclic codes of length ( p r , p s ) over F p m R .