Trivial ring

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

r × r = r
r + r = r

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.

The trivial ring is also sometimes called the zero ring, because {0} is a ring under the standard operations of addition and multiplication.