A module 
over a unit ring 
is called faithful if for all distinct elements 
, 
of 
,
there exists 
such that 
.
In other words, the multiplications by 
and by 
define two different endomorphisms
of 
.
This condition is equivalent to requiring that whenever 
, 
, one has that 
for some 
, i.e., 
, so that the annihilator
of 
is reduced to 
.
This shows, in particular, that any torsion-free module is faithful. Hence the field of rationals 
and the polynomial rings 
are faithful 
-modules.
More generally, any ring 
containing 
as a subring is faithful as a
module over 
, since 1 is annihilated only by 0.
The 
-modules
are not faithful, since they are
annihilated by 
.
In general, a finite module over an infinite ring cannot be faithful, since in this
case the infinitely many elements of the ring have to give rise to only a finite
number of module endomorphisms.
