Zero ring

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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 to refer to any rng of square zero, i.e., a rng in which xy = 0 for all x and y. This article refers to 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[edit]

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[edit]

Constructions[edit]

Notes[edit]

  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.

References[edit]