Rings of polynomials R N = Z p [ x]/ x N − 1 which are isomorphic to Z P N are studied, where p is prime and N is an integer. If I is an ideal in R N , the code K whose vectors constitute` the isomorphic image of I is a linear cyclic code. If I is a principle ideal and K contains only the trivial cycle {0} and one nontrivial cycle of maximal least period N, then the code words of K/{0} obtained by removing the zero vector can be arranged in an order which constitutes a linear circulant matrix, C. The distribution of the elements of C is such that it forms the cyclic core of a generalized Hadamard matrix over the additive group of Z Pp . A necessary condition that C = K/{0} be linear circulant is that for each row vector v of C, the periodic infinite sequence a( v) produced by cycling the elements of v be period invariant under an arbitrary permutation of the elements of the first period. The necessary and sufficient condition that C be linear circulant is that the dual ideal generated by the parity check polynomial h(χ) of K be maximal (a nontrivial, prime ideal of R N ), with N = p k − 1 and k = deg ( h(χ)).