A module M over a unit ring R is called faithful if for all distinct elements a, b of R, there exists x element M such that a x!=b x. In other words, the multiplications by a and by b define two different endomorphisms of M. This condition is equivalent to requiring that whenever a element R, a!=0, one has that a x!=0 for some x element M, i.e., x M!=0, so that the annihilator of M is reduced to {0}. This shows, in particular, that any torsion-free module is faithful. Hence the field of rationals Q and the polynomial rings Z〈X_1, ..., X_n〉 are faithful Z-modules.