FACTOID # 10: The total number of state executions in 2005 was 60: 19 in Texas and 41 elsewhere. The racial split was 19 Black and 41 White.

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Encyclopedia > Trivial ring

A trivial ring is a ring defined on a singleton set, {x}. The ring operations (* and +) are trivial:

x * x = x
x + x = x

Clearly this ring is commutative. Its single element is both the additive and the multiplicative identity element, i.e. x=0=1.

A ring R is trivial if and only if 1 = 0, since this equality implies that for all r within R, r = r * 1 = r * 0 = 0.

Results from FactBites:

 PlanetMath: examples of initial objects and terminal objects and zero objects (616 words) In the category of groups, any trivial group (consisting only of its identity element) is a zero object. In the category of rings with identity, the ring of integers (and any ring isomorphic to it) serves as an initial object. The empty scheme (which is the prime spectrum of the zero ring) is an initial object.
 Ring (mathematics) - Wikipedia, the free encyclopedia (1183 words) In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. The split-complex plane D is a ring useful in modern physics and is a subring of the tessarines.
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