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