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
.