A trivial ring is 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.
