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:
“In our most trivial walks, we are constantly, though unconsciously, steering like pilots by certain well-known beacons and headlands, and if we go beyond our usual course we still carry in our minds the bearing of some neighboring cape; and not till we are completely lost, or turned round,—for a man needs only to be turned round once with his eyes shut in this world to be lost,—do we appreciate the vastness and strangeness of nature.”
—Henry David Thoreau (1817–1862)
“Close friends contribute to our personal growth. They also contribute to our personal pleasure, making the music sound sweeter, the wine taste richer, the laughter ring louder because they are there.”
—Judith Viorst (20th century)