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    Faithful Module

    Definition

    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. More generally, any ring S containing R as a subring is faithful as a module over R, since 1 is annihilated only by 0.

    Related term

    faithfully flat module

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