Polynomial codes and principal ideal rings

Conditions are given which determine when the ring R=S[x/sub 1/, ..., x/sub n/]/(f/sub 1/(x/sub 1/), ..., f/sub n/(x/sub n/)) is a principal ideal ring where either S=Z/sub m/ and R is finite or S is a field. If S and R are both finite then an ideal C of R is a linear code. Hence one obtains necessary and sufficient conditions for the existence of a single generator polynomial of C. If S is a finite field and f/sub i/(x/sub i/)=x/sub i//sup li/-1 for certain integers l/sub i/ for i=1 to n then the ideals of R are multivariate cyclic codes and include the class of generalized Reed-Muller codes.