Trivial ring

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

$r \times r = r \,$

$r + r = r. \,$

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.,

$r = 0 = 1. \,$

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,

$r = r \times 1 = r \times 0 = 0. \,$

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

See also

• Field with one element

References

• David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10. ISBN 0-521-33718-6.