Trivial ring
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a trivial ring is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
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
Zero ring
In ring theory, a branch of mathematics, a zero ring is a ring in which the product of any two elements is 0 . In ring theory, a branch of mathematics, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element). In ring theory, a branch of...
, 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
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
, since the single element is its own multiplicative inverse.