zero vector space
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In algebra, the zero object of a given
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
is, in the sense explained below, the simplest object of such structure. As a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
, and the only element is called zero, so the object itself is typically denoted as . One often refers to ''the'' trivial object (of a specified category) since every trivial object is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to any other (under a unique isomorphism). Instances of the zero object include, but are not limited to the following: * As a group, the zero group or trivial group. * As a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, the zero ring or trivial ring. * As an
algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
or algebra over a ring, the trivial algebra. * As a
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 * Modul ...
(over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
 ), the zero module. The term trivial module is also used, although it may be ambiguous, as a ''trivial G-module'' is a
G-module In mathematics, given a group ''G'', a ''G''-module is an abelian group ''M'' on which ''G'' acts compatibly with the abelian group structure on ''M''. This widely applicable notion generalizes that of a representation of ''G''. Group (co)homo ...
with a trivial action. * As a vector space (over a field ), the zero vector space, zero-dimensional vector space or just zero space. These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties. In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as: : , where . The most general of them, the zero module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, , because there are no non-zero elements. This structure is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
and commutative. A ring which has both an additive and multiplicative identity is trivial if and only if , since this equality implies that for all within , :r = r \times 1 = r \times 0 = 0 . In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of depend on exact definition of the multiplicative identity; see ' below. Any trivial algebra is also a trivial ring. A trivial
algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
is simultaneously a zero vector space considered
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
. Over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
, a trivial algebra is simultaneously a zero module. The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra. The zero-dimensional is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
, and a ''trivial module'' mentioned above.


Properties

The trivial ring, zero module and zero vector space are zero objects of the corresponding categories, namely Rng, -Mod and Vect. The zero object, by definition, must be a terminal object, which means that a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
  must exist and be unique for an arbitrary object . This morphism maps any element of  to . The zero object, also by definition, must be an initial object, which means that a morphism  must exist and be unique for an arbitrary object . This morphism maps , the only element of , to the zero element , called the zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to . For modules and vector spaces, this
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
  is the only empty-generated submodule (or 0-dimensional
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
) in each module (or vector space) .


Unital structures

The object is a terminal object of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an initial object (and hence, a ''zero object'' in the category-theoretical sense) depend on exact definition of the
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
 1 in a specified structure. If the definition of  requires that , then the object cannot exist because it may contain only one element. In particular, the zero ring is not a field. If mathematicians sometimes talk about a field with one element, this abstract and somewhat mysterious mathematical object is not a field. In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the object can exist. But not as initial object because identity-preserving morphisms from to any object where do not exist. For example, in the category of rings Ring the ring of integers Z is the initial object, not . If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor , then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.


Notation

Zero vector spaces and zero modules are usually denoted by (instead of ). This is always the case when they occur in an exact sequence.


See also

*
Nildimensional space In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical i ...
*
Triviality (mathematics) In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...
*
Examples of vector spaces This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis. ''Notation''. Let ''F'' denote an arbitrary field such as the real numbers R or the complex numbers C. T ...
* Field with one element *
Empty semigroup In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a ''non-empty'' set ...
* Zero element *
List of zero terms A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...


External links

* * * {{MathWorld, title=Zero Module, id=ZeroModule, author=Barile, Margherita 0 0
Object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
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