Integral Representations of Cyclic Groups of Prime Order

IRVING REINER11. Elementary facts.In this paper we shall extend a result due to Diederichsen [2 ] on integral representations of cyclic groups of prime order, and shall simplify the proof thereof.Let Z denote the ring of rational integers, Q the rational field.If R is a ring, by a regular Rmodule we shall mean a finitely-generated torsion-free P-module.Lemma 1 (Zassenhaus [9]).Let R be a regular Z-module contained in afield K, and suppose R contains a Q-basis of K. Then every irreducible regular R-module is R-isomorphic to an ideal in R. Two ideals in R are R-isomorphic (as R-modules) if and only if they lie in the same ideal class.