Trivial ring

A trivial ring is a ring defined on a singleton set, {"r"}. The ring operations (× and +) are trivial:

:$r\; imes\; r\; =\; r$:$r\; +\; r\; =\; r\; ,$ Clearly this ring is commutative. Its single element is both the additive and the multiplicative identity element, i.e., :$r\; =\; 0\; =\; 1\; ,$

A ring "R" is trivial if and only if 1 = 0, since this equality implies that for all "r" within "R", :$r\; =\; r\; imes\; 1\; =\; r\; imes\; 0\; =\; 0$

Any two trivial rings are isomorphic, so one may also speak about "the" trivial ring.

The trivial ring is also sometimes called the zero ring, because {0} (where 0 here corresponds to the number 0) is a ring under the standard operations of addition and multiplication.

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