Zero ring

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.

• 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 also denoted Z₁.
• 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/nZ (or Z_n, which is isomorphic to Z/nZ) 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.

• 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⁻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.

• 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.