CF-Permutation modules

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Let M be a free module of finite rank over a commutative ring with unity R. Let R[X] be the polynomial ring with coefficients in R. For an R-endomorphism f of M and a polynomial P(X) of R[X] and under certain condition, we show that if the R[X]-module MP(f) defined via P(f) is a CF-module, then the R[X]-module Mf defined via f is also a CF-module. An application to modules over group rings is also given. For a prime number p and a finite group G, we characterize CF-permutation G-modules.