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trivial ring

Contents

Contents

Definition

The trivial ring, or zero ring, is the ring with a single element, which is both 00 and 11.

One usually denote the trivial ring as 00 or {0}\{0\}, even though 11 or {1}\{1\} would make as much sense.

Properties

  • The trivial ring is the only ring in which 0=10 = 1, since if that is the case then it follows for all elements xx that x=1x=0x=0x = 1 x = 0 x = 0.

  • The trivial ring is the terminal object in Rings. It is both terminal and initial (hence a zero object) in the category of nonunital rings, but it is not initial in Rings (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, hence it is a strict terminal object there.

  • The trivial ring is an example of a trivial algebra.

Last revised on April 16, 2021 at 03:41:33. See the history of this page for a list of all contributions to it.