Cyclic codes and minimal strong Gröbner bases over a principal ideal ring

We characterise minimal strong Gröbner bases of R[x] where R is a commutative principal ideal ring and deduce a structure theorem for cyclic codes of arbitary length over R. When R is an Artinian chain ring with residue field R and gcd(char(R),n) = 1, we recover a theorem for cyclic codes of length n over R due to Calderbank and Sloane for R = Z/pkZ.