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In
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the zero ring or trivial ring is the unique ring (up to
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
, the zero ring is the
terminal object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, whereas the ring of integers Z is the
initial object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
.


Definition

The zero ring, denoted or simply 0, consists of the one-element set with the
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
+ and · defined such that 0 + 0 = 0 and 0 · 0 = 0.


Properties

* The zero ring is the unique ring in which the
additive identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
0 and
multiplicative identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies Here Technologies (trading as A trade name, trading name, or business name is a pseudonym A pseudonym () or alias () (originally: ...
.) * The zero ring is commutative. * The element 0 in the zero ring is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
, serving as its own
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

multiplicative inverse
. * The
unit group In the branch of abstract algebra known as ring theory, a unit of a ring (mathematics), ring R is any element u \in R that has a multiplicative inverse in R: an element v \in R such that :vu = uv = 1, where is the multiplicative identity. The s ...
of the zero ring is the
trivial group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. * The element 0 in the zero ring is not a
zero divisor In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
. * The only
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. * The zero ring is not a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
; 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
", 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. Whether the zero ring is considered to be a
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
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 In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical an ...
from ''A'' to the zero ring. Thus the zero ring is a
terminal object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
in the
category of rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
. * 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of any nonzero ring. * The zero ring is the unique ring of
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
1. * The only
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
. It is, however, a semilocal ring. * The zero ring is Artinian and (therefore)
NoetherianIn mathematics, the adjective In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that grammatical modifier, modifies a noun or noun phrase or describes its referent. Its Semantics, semantic role is to change inf ...
. * The
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models expla ...
of the zero ring is the empty scheme.Hartshorne, p. 80. * The
Krull dimension In commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with ...
of the zero ring is −∞. * The zero ring is semisimple but not
simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
. * The zero ring is not a
central simple algebra In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
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