McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f (X) is an element of R[X] and m(X) is an element of M[X], f (X) m(X) = 0 implies there exists a nonzero r is an element of R (resp., m is an element of M) with rm(X) = 0 (resp., f (X) m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.