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 ring of square zero, i.e., a ring 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 so 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 = 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] 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

• 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. iff I is the unit ideal.
• For any commutative ring A and multiplicative set S in A, the localization S^−1A is the zero ring if and only if S contains 0.
• If A is any ring, then the ring M₀(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

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

• 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.