Let R be a commutative ring with identity and \({\mathbb{N}}\) the set of nonnegative integers. The double sequence \(S=(S_{ij}): {\mathbb{N}}\times {\mathbb{N}}\to R\) is said to be doubly periodic if there exist positive integers p and q such that \(S_{i+p,j}=S_{ij}=S_{i,j+q}\) for all i,j\(\in {\mathbb{N}}\). A number of properties and characterizations of such sequences are given using power series rings in two variables. Conditions for the factorization of such sequences as tensor products of single sequences are given. Applications of these results to automata and product codes are given and conditions determined under which certain sets of doubly periodic sequences can be endowed with a field structure.