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 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,
In this case it is possible to define division by zero, since the single element is its own multiplicative inverse.
It should be emphasized that the trivial ring is not a field and that a field has at least two elements. If mathematicians talk sometimes of a field with one element, this abstract and somewhat mysterious mathematical object is not a set and, in particular, is not a singleton where 1 = 0 is the only element.
Famous quotes containing the words trivial and/or ring:
“Each truth that a writer acquires is a lantern, which he turns full on what facts and thoughts lay already in his mind, and behold, all the mats and rubbish which had littered his garret become precious. Every trivial fact in his private biography becomes an illustration of this new principle, revisits the day, and delights all men by its piquancy and new charm.”
—Ralph Waldo Emerson (1803–1882)
“Rich and rare were the gems she wore,
And a bright gold ring on her hand she bore.”
—Thomas Moore (1779–1852)