A ring defined on a singleton set 
.
The ring operations (multiplication and addition) are defined in the only possible
way,
 |
(1)
|
and
 |
(2)
|
It follows that this is a commutative unit ring, where 
is the multiplicative
identity. Of course, 
also coincides with the additive identity, i.e.,
it is the so-called zero element of the ring.
For this reason, the trivial ring is often denoted 
and also called the zero ring. In fact, the subset 
is the only trivial subring of the
ring of integers 
.
A unit ring 
is trivial whenever 
,
since this equality implies that for all 
 |
(3)
|
A trivial ring is a trivial module over itself.
