# Zero ring

(Redirected from Trivial ring)

In ring theory, a branch of mathematics, the zero ring[1][2][3][4][5] or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used more generally to refer to any pseudo-ring of square zero, i.e., a pseudo-ring in which xy = 0 for all x and y. But this article is about the one-element ring.)

In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object.

## Definition

The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and · defined so that 0 + 0 = 0 and 0 · 0 = 0.

## Properties

$1=0 \iff \forall r \in R: r = r \times 1 = r \times 0 = 0. \,$

## Notes

1. ^ Artin, p. 347.
2. ^ Atiyah and Macdonald, p. 1.
3. ^ Bosch, p. 10.
4. ^ Bourbaki, p. 101.
5. ^ Lam, p. 1.
6. ^ Artin, p. 347.
7. ^ Lang, p. 83.
8. ^ Bosch, p. 10.
9. ^ Hartshorne, p. 80.
10. ^ Hartshorne, p. 80.
11. ^ Hartshorne, p. 80.