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]

  • The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide.[6][7] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0.)
  • The zero ring is commutative.
  • The element 0 in the zero ring is a unit, serving as its own multiplicative inverse.
  • The unit group of the zero ring is the trivial group {0}.
  • The element 0 in the zero ring is not a zero divisor.
  • The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither maximal nor prime.
  • The zero ring is not a field; this agrees with the fact that its zero ideal is not maximal. In fact, there is no field with less than 2 elements. (When mathematicians speak of the "field with one element", they are referring to a non-existent object, and their intention is to define the category that would be the category of schemes over this object if it existed.)
  • The zero ring is not an integral domain;[8] this agrees with the fact that its zero ideal is not prime. Whether the zero ring is considered to be a domain at all is a matter of convention, but there are two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor (in particular, 0 is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer n, the ring Z/nZ is a domain if and only if n is prime.
  • For each ring A, there is a unique ring homomorphism from A to the zero ring. Thus the zero ring is a terminal object in the category of rings.[9]
  • If A is a nonzero ring, then there is no ring homomorphism from the zero ring to A. In particular, the zero ring is not a subring of any nonzero ring.[10]
  • The characteristic of the zero ring is 1.
  • The only module for the zero ring is the zero module. It is free of rank א for any cardinal number א.
  • The zero ring is not a local ring. It is, however, a semilocal ring.
  • The zero ring is Artinian and (therefore) Noetherian.
  • The spectrum of the zero ring is the empty scheme.[11]
  • The Krull dimension of the zero ring is −∞.
  • The zero ring is semisimple but not simple.
  • The zero ring is not a central simple algebra over any field.
  • The total quotient ring of the zero ring is itself.

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. ^ Lam, p. 3.
  9. ^ Hartshorne, p. 80.
  10. ^ Hartshorne, p. 80.
  11. ^ Hartshorne, p. 80.

References[edit]