Trivial Ring

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

One often refers to the trivial ring since every trivial ring is isomorphic to any other (under a unique isomorphism). The element of the trivial ring is usually chosen to be the number 0, because {0} is a ring under the standard operations of addition and multiplication. For this reason, it is often called the zero ring (not to be confused with a zero ring, although the trivial ring is a zero ring).

Clearly the trivial ring is commutative. Its single element is both the additive and the multiplicative identity element, i.e.,

A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R,

In this case it is possible to define division by zero, since the single element is its own multiplicative inverse.

It should be emphasized that the trivial ring is not a field and that a field has at least two elements. If mathematicians talk sometimes of a field with one element, this abstract and somewhat mysterious mathematical object is not a set and, in particular, is not a singleton where 1 = 0 is the only element.