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Zero ring
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Not to be confused with ring 0.
Algebraic structure → Ring theory
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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

[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.[1][6] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. 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 {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 generally excluded from fields, while occasionally called as the trivial field. Excluding it agrees with the fact that its zero ideal is not maximal. (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 generally excluded from integral domains.[7] 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, 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.[8]
  • 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.[8]
  • 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.[8]
  • 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]

Citations

[edit]
  1. ^ a b Artin 1991, p. 347
  2. ^ Atiyah & Macdonald 1969, p. 1
  3. ^ Bosch 2012, p. 10
  4. ^ Bourbaki, p. 101
  5. ^ Lam 2003, p. 1
  6. ^ Lang 2002, p. 83
  7. ^ Lam 2003, p. 3
  8. ^ a b c Hartshorne 1977, p. 80

References

[edit]
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Zero ring

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In , the zero ring, also known as the trivial ring, is the unique ring consisting of a single element, denoted 0, which functions as both the and the multiplicative identity, with the operations defined by 0+0=00 + 0 = 0 and 0⋅0=00 \cdot 0 = 0. This structure forms a where the unity element 1 coincides with 0, implying that every element in the ring is zero. Although there are infinitely many isomorphic copies of the zero ring, they are all trivially equivalent under ring isomorphisms. A key property of the zero ring is that it satisfies the ring axioms in a degenerate manner: for any element xx (which must be 0), x=1⋅x=0⋅x=0x = 1 \cdot x = 0 \cdot x = 0, confirming its trivial nature. It serves as the terminal object in the , meaning there exists a unique from any ring to the zero ring, mapping all elements to 0. However, it is not an initial object, as the Z\mathbb{Z} fulfills that role instead. The zero ring is also a trivial module over itself and appears as the only trivial subring of Z\mathbb{Z}. Notably, the zero ring does not qualify as a field or an , as these structures require the additive and multiplicative identities to be distinct (i.e., 0≠10 \neq 1).

Core Concepts

Definition

The zero ring, also known as the trivial ring, is the unique ring (up to ) consisting of a single element denoted by 0, equipped with binary operations of and defined by 0+0=00 + 0 = 0 and 0⋅0=00 \cdot 0 = 0. This structure forms an algebraic system where the sole element serves as both the additive and multiplicative identity, though the latter is degenerate in the sense that it coincides with the additive zero. This construction satisfies the standard ring axioms without requiring a multiplicative identity distinct from the additive zero. Specifically, the additive group ({[0](/page/0)},+)( \{[0](/page/0)\}, + ) is the trivial , fulfilling closure, associativity, commutativity, the existence of the identity 0, and additive inverses (since −0=0-0 = 0). Multiplication is associative, as (0⋅0)⋅0=0=0⋅(0⋅0)(0 \cdot 0) \cdot 0 = 0 = 0 \cdot (0 \cdot 0), and the distributive laws hold trivially: for all a,b,c∈{0}a, b, c \in \{0\}, a⋅(b+c)=0=0+0=a⋅b+a⋅ca \cdot (b + c) = 0 = 0 + 0 = a \cdot b + a \cdot c, and similarly for the other distributivity axiom. The zero ring is commonly denoted by {0}\{0\} or simply as the trivial ring to emphasize its minimal nature. It must be distinguished from the zero ideal {0}\{0\} in a nonzero ring, which is a proper rather than the entire ring structure. The zero ring was first implicitly considered in early 20th-century texts, such as Abraham Fraenkel's axiomatic treatment of rings, where it was explicitly excluded to focus on systems with regular elements.

Elementary Properties

In the zero ring, the additive identity coincides with the multiplicative identity, so 0=10 = 1. This equality follows directly from the ring's structure as the singleton set {0}\{0\} equipped with the trivial operations 0+0=00 + 0 = 0 and 0⋅0=00 \cdot 0 = 0. The single element 00 serves as its own , since 0+0=00 + 0 = 0, satisfying the requirement that −0+0=0-0 + 0 = 0. Similarly, 00 is its own when inverses are considered, as 0⋅0=0=10 \cdot 0 = 0 = 1. The zero ring is commutative under both addition and multiplication by triviality, as the only possible products and sums are 0+0=00 + 0 = 0 and 0⋅0=00 \cdot 0 = 0, which equal their reverses. The characteristic of the zero ring is 11, the smallest positive integer nn such that n⋅0=0n \cdot 0 = 0 for all elements, specifically via 1⋅0=01 \cdot 0 = 0; this is atypical, as nonzero rings have characteristic 00 or a prime number at least 22. Up to , there is a unique , as it is the only ring in which 1=01 = 0, and any singleton set with these trivial operations is isomorphic via the identity map.

Algebraic Structure

Operations and Identities

The , denoted as {0}, consists of a single element that serves as both the additive and multiplicative identity, with operations defined in the only possible way consistent with ring axioms. Addition is the operation where 0+0=00 + 0 = 0, forming an of order one. Multiplication is similarly trivial, with 0⋅0=00 \cdot 0 = 0, and distributes over vacuously since 0⋅(0+0)=0⋅0=0=0+00 \cdot (0 + 0) = 0 \cdot 0 = 0 = 0 + 0. These operations illustrate the complete triviality of the structure, as shown in the following Cayley tables:

Addition Table

++0
00

Multiplication Table

⋅\cdot0
00
The coincidence of identities in the zero ring arises because the additive identity 0 must function as the multiplicative identity. In any ring with a multiplicative identity 1, the property 1⋅r=r1 \cdot r = r holds for all rr; substituting r=0r = 0 gives 1⋅0=01 \cdot 0 = 0. However, since 0⋅r=00 \cdot r = 0 for all rr in any ring (by the absorbing property derived from distributivity: 0⋅r=(0+0)⋅r=0⋅r+0⋅r0 \cdot r = (0 + 0) \cdot r = 0 \cdot r + 0 \cdot r, implying 0⋅r=00 \cdot r = 0 by additive cancellation), if 1 = 0, then 0⋅r=r0 \cdot r = r forces r=0r = 0 for all rr, confirming the ring has only one element. Thus, in the zero ring, 0 satisfies 0⋅x=x⋅0=x0 \cdot x = x \cdot 0 = x trivially for the sole element x=0x = 0. Regarding zero divisors, an element a≠0a \neq 0 is a if there exists b≠0b \neq 0 such that a⋅b=0a \cdot b = 0. In the zero ring, no nonzero elements exist, so the set of zero divisors is empty. Nonetheless, the structure implies that any nonzero element (of which there are none) would be a zero divisor, as multiplication by 0 yields 0 for any bb. This vacuous situation underscores the degenerate nature of the ring. The element 0 in the zero ring exhibits the absorptive property, annihilating every element under : 0⋅x=x⋅0=00 \cdot x = x \cdot 0 = 0 for all x∈{0}x \in \{0\}, which holds by the definition of the operations. This property extends the general ring behavior where 0 absorbs all products, but here it is entirely self-contained.

Ideals and Modules

In the zero ring, denoted R={0}R = \{0\}, the only ideal is RR itself, as there are no proper subsets that satisfy the ideal axioms beyond the entire ring. The zero ring has no proper ideals. In particular, it has no prime or , leading to an empty Spec⁡(R)=∅\operatorname{Spec}(R) = \emptyset. The of the zero ring is often taken to be −∞-\infty, reflecting the absence of any chain of , as the supremum over the of chain lengths is conventionally negative infinity in this context. Conventions for the of the zero ring vary across texts, with values such as −∞-\infty, -1, or even 0 or 1 for convenience in certain proofs. Regarding chain conditions, the zero ring satisfies both the ascending and descending chain conditions on ideals (ACC and DCC), making it Noetherian and Artinian, since the only possible ideal chain is the stationary sequence involving {0}\{0\}, which stabilizes immediately. It is also semilocal, with the single (improper) {0}\{0\}, as there are no other ideals to form additional maximal ones. Turning to modules, the category of RR-modules over the zero ring is trivial, consisting solely of the zero module {0}\{0\}, because any module MM must satisfy 1⋅m=m1 \cdot m = m for all m∈Mm \in M, but since 1=01 = 0 in RR, this implies 0⋅m=0=m0 \cdot m = 0 = m, forcing M={0}M = \{0\} with the zero action. No nontrivial modules exist, as scalar multiplication by any ring element (which is 0) always yields the zero vector, incompatible with a nonzero underlying unless the group is trivial. This triviality underscores the degenerate nature of module theory over the zero ring, where all homomorphisms and extensions collapse to the zero map.

Categorical Role

Homomorphisms

In the , the zero ring {[0](/page/0)}\{[0](/page/0)\} serves as object, characterized by the universal property that for any ring RR, there exists a unique ϕ:R→{0}\phi: R \to \{0\}. This homomorphism is the constant zero , which sends every element of RR to the sole element 0 in {0}\{0\}. It preserves the ring operations since ϕ(r+s)=0=0+0=ϕ(r)+ϕ(s)\phi(r + s) = 0 = 0 + 0 = \phi(r) + \phi(s) and ϕ(rs)=0=0⋅0=ϕ(r)ϕ(s)\phi(rs) = 0 = 0 \cdot 0 = \phi(r) \phi(s) for all r,s∈Rr, s \in R. The uniqueness of this homomorphism follows from the requirement that ring homomorphisms preserve the multiplicative identity (in the unital setting). Specifically, ϕ(1R)=1{0}=[0](/page/0)\phi(1_R) = 1_{\{0\}} = [0](/page/0), and for any r∈Rr \in R, ϕ(r)=ϕ(r⋅1R)=ϕ(r)⋅ϕ(1R)=ϕ(r)⋅[0](/page/0)=[0](/page/0)\phi(r) = \phi(r \cdot 1_R) = \phi(r) \cdot \phi(1_R) = \phi(r) \cdot [0](/page/0) = [0](/page/0). Thus, any such ϕ\phi must be the zero map. Regarding homomorphisms out of the zero ring, there are no nontrivial ring homomorphisms from {[0](/page/0)}\{[0](/page/0)\} to a nonzero ring SS, except the zero map itself, which sends to in SS. In the unital category, even the zero map fails to preserve the identity unless SS is also the zero ring, as it would require 1S=ϕ(1{[0](/page/0)})=ϕ([0](/page/0))=[0](/page/0)1_S = \phi(1_{\{[0](/page/0)\}}) = \phi([0](/page/0)) = [0](/page/0). Under some definitions of rings without unity (or where homomorphisms do not preserve unity), the zero ring acts as an initial object in the , with the zero map providing the unique to any other ring; however, it is primarily recognized as terminal in standard unital .

Terminal Object

In the , denoted Ring, the zero ring serves as the terminal object. This means that for every ring RR, there exists exactly one from RR to the zero ring, which maps every element of RR to the single element 00 of the zero ring. This terminal property holds regardless of whether rings are required to have a multiplicative identity (unital rings) or not, as in the category of rngs (rings without unity). In the category of rngs, the zero ring remains terminal, with the unique from any rng SS sending all elements to 00, preserving the additive and multiplicative structures. Regarding limits in Ring, the zero ring's role as terminal object implies that products involving it are trivial: the product of any ring with the zero ring is isomorphic to the zero ring itself, as the projection maps enforce the zero structure. , however, exhibit greater complexity; the coproduct of the zero ring with a nonzero ring TT is isomorphic to TT, but direct sums or free products in noncommutative cases require careful construction to handle the zero component. In contrast, the zero ring does not serve as a terminal object in stricter categories like that of fields (Field) or integral domains, where it is excluded because it lacks the necessary nonzero multiplicative identity or zero divisors. Dually, in the opposite category Ringop, the zero ring functions as the object, highlighting its symmetric yet distinct roles across categorical duals.

Theoretical Debates

Inclusion in Ring Theory

The debate over the inclusion of the in centers on varying definitional criteria for rings, particularly the requirement for a multiplicative identity. Early 20th-century definitions, such as Adolf Fraenkel's 1914 axiomatization, which mandated a multiplicative identity, and the 1916 requiring at least one regular (non-zero-divisor) element, explicitly excluding the zero ring to avoid trivial cases. In contrast, Emmy Noether's 1921 of commutative rings did not require a multiplicative identity, allowing structures like the zero ring as a limiting case. The Bourbaki group's initial 1958 treatment in Algèbre followed this non-unital approach, but their 1970 revision adopted a unital definition, implicitly excluding the zero ring by assuming a unit element distinct from zero. In modern , the zero ring is often included under general definitions but treated with caveats due to its pathological behavior. Many influential texts explicitly exclude it; for instance, Atiyah and Macdonald () define rings as commutative with a unit 1 and state that every ring A≠[0](/page/0)A \neq [0](/page/0) has a , thereby omitting the zero ring from core results. This consensus reflects a preference for non-trivial structures in algebraic developments, where the zero ring disrupts standard theorems without providing substantial insights. However, broader treatments, such as those in non-commutative or general algebra, may permit it as a rng (ring without identity) with the element serving as a vacuous unity. The zero ring's potential unity arises from the coincidence of additive and multiplicative identities, where acts as 1, satisfying the axioms 1⋅r=r⋅1=r1 \cdot r = r \cdot 1 = r for all rr (trivially, since the only element is ). This vacuous fulfillment aligns with minimal ring axioms but introduces inconsistencies in theorems presupposing 1≠01 \neq 0, such as the existence of maximal ideals or properties of units. For example, assuming 1≠01 \neq 0 ensures non-degenerate homomorphisms and module structures, avoiding collapses to the trivial case. Post-1970s developments, influenced by category-theoretic perspectives, have increasingly incorporated the for structural completeness, viewing it as an extremal object in categories of rings despite its exclusion from unital subclasses. This evolution prioritizes categorical universality over strict definitional exclusion, allowing the zero ring in foundational contexts while noting its limited utility in applied theory.

Exclusion from Ring Classes

The zero ring fails to qualify as a field because it violates the standard axiom requiring the 0 and the multiplicative identity 1 to be distinct elements. In the zero ring, the single element serves as both 0 and 1, rendering the trivial and incompatible with the requirement that every nonzero element possess a . Consequently, conventions in explicitly exclude the zero ring from the class of fields to maintain the nontrivial nature of these structures. Similarly, the zero ring is not an , as definitions of integral domains mandate a with unity where 1 ≠ 0 and no nonzero zero divisors exist. Although the zero ring has no nonzero elements to serve as zero divisors, the collapse of 0 and 1 undermines the foundational distinction required for domains, leading to its exclusion by convention. This ensures that integral domains support meaningful notions of integrality and without degenerating into triviality. The zero ring also cannot be classified as a division ring (or skew field), where every nonzero element must be invertible, because its sole element equates the identities, precluding the existence of nonzero elements altogether. Euclidean domains, as a subclass of domains equipped with a Euclidean function for division algorithms, inherit these exclusionary criteria due to the absence of a proper unity and the characteristic 1 property that bars normed structures. These omissions preserve the algebraic utility of such classes for applications in and . In broader , the zero ring is frequently omitted from theorem statements to avoid trivial or failed cases; for instance, results on prime or maximal ideals often specify "nonzero rings" explicitly, as the zero ring's sole ideal coincides with itself, rendering concepts like fields or residue classes meaningless. Such exclusions ensure theorems apply nontrivially, as including the zero ring would either vacuously hold or contradict expected properties, such as the equivalence of maximal ideals with fields in quotients. One rare context where the zero ring appears is in , particularly scheme theory, where its spectrum corresponds to the empty scheme, representing the initial object in the category of schemes; however, even here, it is handled separately to avoid complications in sheaf-theoretic constructions. This inclusion highlights the zero ring's role as a boundary case but reinforces its general exclusion from standard ring subclasses.

References

  1. https://mathworld.wolfram.com/TrivialRing.html
  2. https://artofproblemsolving.com/wiki/index.php/Zero_ring
  3. https://www.maths.gla.ac.uk/~jnicholson/algebra3-notes.pdf
  4. https://kconrad.math.uconn.edu/blurbs/ringtheory/ringdefs.pdf
  5. https://projecteuclid.org/journals/modern-logic/volume-8/issue-1-2/The-origins-of-the-definition-of-abstract-rings/rml/1081878062.pdf
  6. https://www.math.cuhk.edu.hk/course_builder/1920/math3030/note_ring.pdf
  7. https://mathweb.ucsd.edu/~drogalsk/200-coursenotes-2021-03-12.pdf
  8. https://www.math.columbia.edu/~khovanov/MA2_2022/files/01_rings.pdf
  9. https://math.arizona.edu/~cais/594Page/soln/soln6.pdf
  10. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_(Clark)/01%3A_Chapters/1.09%3A_Introduction_to_Ring_Theory
  11. https://people.tamu.edu/~yvorobets/MATH433-2023A/Lect2-10web.pdf
  12. https://www.jmilne.org/math/xnotes/CA402.pdf
  13. https://www.dpmms.cam.ac.uk/~sjw47/Handout.pdf
  14. https://uregina.ca/~franklam/Math420_F2022_Notes.pdf
  15. https://ncatlab.org/nlab/show/zero%2Bobject
  16. https://ncatlab.org/nlab/show/terminal%2Bobject
  17. https://stacks.math.columbia.edu/tag/0006
  18. https://www.johndcook.com/blog/initial-final-objects/
  19. https://math.mit.edu/~poonen/papers/ring.pdf
  20. https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_lect8.pdf
  21. https://mathweb.ucsd.edu/~drogalsk/200-coursenotes-2021-03-03.pdf
  22. https://www.math.utah.edu/~schwede/math6320/CANotes.pdf
  23. https://kconrad.math.uconn.edu/blurbs/ringtheory/ideals.pdf
  24. https://www.math.lsu.edu/~madden/M7210fall2012/lec20121026-Units-ZeroDiv.pdf
  25. http://danaernst.com/teaching/mat411s16/Rings.pdf
  26. https://www.memphis.edu/msci/people/pbalistr/rings.pdf
  27. https://stacks.math.columbia.edu/tag/00DY
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