A faithful module is a module 
over a unit ring 
such that 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.
