Zero ring

From Wikipedia, the free encyclopedia

  (Redirected from Trivial ring)

Jump to navigation Jump to search

Algebraic structure → Ring theory Ring theory
Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total quotient ring • Product ring • Free product of associative algebras • Tensor product of rings Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring ${\displaystyle \mathbb {Z} }$ • Terminal ring ${\displaystyle 0=\mathbb {Z} _{1}}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring ${\displaystyle \mathbb {Z} _{1}}$ • Integers modulo pn ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ • Prüfer p-ring ${\displaystyle \mathbb {Z} (p^{\infty })}$ • Base-p circle ring ${\displaystyle \mathbb {T} }$ • Base-p integers ${\displaystyle \mathbb {Z} }$ • p-adic rationals ${\displaystyle \mathbb {Z} [1/p]}$ • Base-p real numbers ${\displaystyle \mathbb {R} }$ • p-adic integers ${\displaystyle \mathbb {Z} _{p}}$ • p-adic numbers ${\displaystyle \mathbb {Q} _{p}}$ • p-adic solenoid ${\displaystyle \mathbb {T} _{p}}$ Algebraic geometry • Affine variety
Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
v t e

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.

Contents

• 1 Definition
• 2 Properties
• 3 Constructions
• 4 Notes
• 5 References

Definition[edit]

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[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 also denoted Z₁.^{[citation needed]}
• 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 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.^[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]

• 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.

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]

• 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.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Zero_ring&oldid=988284639"

Categories:

• Ring theory
• Finite rings

Hidden categories:

• All articles with unsourced statements
• Articles with unsourced statements from November 2020

Navigation menu

Personal tools

• Not logged in
• Talk
• Contributions
• Create account
• Log in

Namespaces

• Article
• Talk

Variants

Views

• Read
• Edit
• View history

More

Search

Navigation

• Main page
• Contents
• Current events
• Random article
• About Wikipedia
• Contact us
• Donate

Contribute

• Help
• Learn to edit
• Community portal
• Recent changes
• Upload file

Tools

• What links here
• Related changes
• Upload file
• Special pages
• Permanent link
• Page information
• Cite this page
• Wikidata item

Print/export

• Download as PDF
• Printable version

Languages

• العربية
• Català
• Deutsch
• Español
• Français
• 한국어
• Nederlands
• 日本語
• Polski

Edit links

• This page was last edited on 12 November 2020, at 06:39 (UTC).
• Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

• Privacy policy
• About Wikipedia
• Disclaimers
• Contact Wikipedia
• Mobile view
• Developers
• Statistics
• Cookie statement

• 
•