trivial ring

The trivial ring, or zero ring, is the ring with a single element, which is both 00 and 11. We usually denote the trivial ring as 00 or {0}\{0\}, even though 11 or {1}\{1\} would make as much sense. It is the only ring in which 0=10 = 1, by the proof

x=1x=0x=0. x = 1 x = 0 x = 0 .

The trivial ring is the terminal object in Ring. It is both terminal and initial (hence a zero object) in the category of nonunital rings, but it is not initial in RingRing itself (defined as the category of unital rings and unital ring homomorphisms). In fact, there are no unital ring homomorphisms from the trivial ring to any nontrivial ring!

The trivial ring is an example of a trivial algebra.

Last revised on September 7, 2010 at 19:11:37. See the history of this page for a list of all contributions to it.