Every module over a ring 
contains a so-called "zero element"
which fulfils the properties suggested by its name with respect to addition,
 |
and with respect to multiplication by any element 
of 
,
 |
This shows that the set 
is closed under both module operations, and, therefore, it itself is a module,
called the zero module. It also deserves the name trivial module, since it is the
simplest module possible.
