A module having only one element: the singleton set 
.
It is a module over any ring 
with respect to the multiplication defined
by
 |
(1)
|
for every 
,
and the addition
 |
(2)
|
which makes it a trivial additive group. The only element 
is, in particular, its zero element.
Therefore, a trivial module is often called the zero module,
and written as 
.
The notion of trivial module is a special case of the more general notion of trivial module structure, which can be defined on every additive Abelian
group 
with respect to every ring 
by setting
 |
(3)
|
for all 
and all 
.
