Trivial ring

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

x * x = x

x + x = x

Clearly this ring is commutative. Its single element is both the additive and the multiplicative identity element, i.e. x=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 * 1 = r * 0 = 0.