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 is trivial if and only if 1 = 0, since this equality implies that for all r within R,
See also
References
- David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10. ISBN 0-521-33718-6.
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