trivial ring

In ring theory, a branch of mathematics, the zero ring 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 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 or simply 0, consists of the one-element set 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. (Proof: If in a ring ''R'', then for all ''r'' in ''R'', we have . The proof of the last equality is found here.)
* 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 .
* The element 0 in the zero ring is not a zero divisor.
* The only ideal in the zero ring is the zero ideal , 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. 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 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.
* 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.
* The zero ring is the unique ring of 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.Hartshorne, p. 80.
* 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 ring, 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 of a ring, localization ''S''⁻¹''A'' is the zero ring if and only if ''S'' contains 0.
* If ''A'' is any ring, then the ring M₀(''A'') of 0 × 0 matrix (mathematics), 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 function, continuous real-valued functions on the empty topological space is the zero ring.

Notes

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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.

Ring theory
Finite rings