# Zero 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 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

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

## Properties

- 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*= 1*r*= 0*r*= 0.) - The zero ring is also denoted
**Z**_{1}. - 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 fewer 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] 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**/*n***Z**(or**Z**_{n}, which is isomorphic to**Z**/*n***Z**) is a domain if and only if*n*is prime, but 1 is not 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 zero ring is by definition the unique ring with the characteristic 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

- For any ring
*A*and ideal*I*of*A*, the quotient*A*/*I*is the zero ring if and only if*I*=*A*, i.e. if and only if*I*is the unit ideal. - For any commutative ring
*A*and multiplicative set*S*in*A*, the localization*S*^{−1}*A*is the zero ring if and only if*S*contains 0. - If
*A*is any ring, then the ring M_{0}(*A*) of 0 × 0 matrices over*A*is the zero ring. - The direct product of an empty collection of rings is the zero ring.
- The endomorphism ring of the trivial group is the zero ring.
- The ring of continuous real-valued functions on the empty topological space is the zero ring.

## Notes

- Artin, p. 347.
- Atiyah and Macdonald, p. 1.
- Bosch, p. 10.
- Bourbaki, p. 101.
- Lam, p. 1.
- Artin, p. 347.
- Lang, p. 83.
- Lam, p. 3.
- Hartshorne, p. 80.
- Hartshorne, p. 80.
- Hartshorne, p. 80.

## References

- Michael Artin,
*Algebra*, Prentice-Hall, 1991. - Siegfried Bosch,
*Algebraic geometry and commutative algebra*, Springer, 2012. - M. F. Atiyah and I. G. Macdonald,
*Introduction to commutative algebra*, Addison-Wesley, 1969. - N. Bourbaki,
*Algebra I, Chapters 1-3*. - Robin Hartshorne,
*Algebraic geometry*, Springer, 1977. - T. Y. Lam,
*Exercises in classical ring theory*, Springer, 2003. - Serge Lang,
*Algebra*3rd ed., Springer, 2002.

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