## Table of Contents

46 relations: Abelian group, Absorbing element, Addition, Additive identity, Algebraic structure, Cartesian product, Category (mathematics), Category of groups, Coproduct, Empty set, Empty sum, Euclidean vector, Field (mathematics), Function composition, Greatest element and least element, Ideal (ring theory), Identity element, Initial and terminal objects, Integer, Lattice (order), Linear algebra, Linear map, Mathematics, Matrix (mathematics), Matrix ring, Module (mathematics), Monoid, Multiplication, Normed vector space, Null semigroup, Partially ordered set, Pointwise, Principal ideal, Product (category theory), Ring (mathematics), Semigroup, Semiring, Tensor, Tensor product, Trivial group, Undergraduate Texts in Mathematics, Union (set theory), Zero divisor, Zero morphism, Zero of a function, 0.

- 0 (number)

## Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

See Zero element and Abelian group

## Absorbing element

In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set.

See Zero element and Absorbing element

## Addition

Addition (usually signified by the plus symbol) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

## Additive identity

In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields. Zero element and additive identity are 0 (number).

See Zero element and Additive identity

## 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 identities, known as axioms, that these operations must satisfy.

See Zero element and Algebraic structure

## Cartesian product

In mathematics, specifically set theory, the Cartesian product of two sets and, denoted, is the set of all ordered pairs where is in and is in.

See Zero element and Cartesian product

## Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

See Zero element and Category (mathematics)

## Category of groups

In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms.

See Zero element and Category of groups

## Coproduct

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.

See Zero element and Coproduct

## Empty set

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

See Zero element and Empty set

## Empty sum

In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. Zero element and empty sum are 0 (number).

See Zero element and Empty sum

## Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction.

See Zero element and Euclidean vector

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

See Zero element and Field (mathematics)

## Function composition

In mathematics, function composition is an operation that takes two functions and, and produces a function such that.

See Zero element and Function composition

## Greatest element and least element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S.

See Zero element and Greatest element and least element

## Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements.

See Zero element and Ideal (ring theory)

## Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied.

See Zero element and Identity element

## Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in, there exists precisely one morphism.

See Zero element and Initial and terminal objects

## Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

## Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

See Zero element and Lattice (order)

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices.

See Zero element and Linear algebra

## Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.

See Zero element and Linear map

## Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Zero element and Mathematics

## Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

See Zero element and Matrix (mathematics)

## Matrix ring

In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication.

See Zero element and Matrix ring

## Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring.

See Zero element and Module (mathematics)

## Monoid

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.

## Multiplication

Multiplication (often denoted by the cross symbol, by the mid-line dot operator, by juxtaposition, or, on computers, by an asterisk) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division.

See Zero element and Multiplication

## Normed vector space

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined.

See Zero element and Normed vector space

## Null semigroup

In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero.

See Zero element and Null semigroup

## Partially ordered set

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.

See Zero element and Partially ordered set

## Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition.

See Zero element and Pointwise

## Principal ideal

In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P.

See Zero element and Principal ideal

## Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

See Zero element and Product (category theory)

## Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

See Zero element and Ring (mathematics)

## Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.

See Zero element and Semigroup

## Semiring

In abstract algebra, a semiring is an algebraic structure.

## Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space.

## Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.

See Zero element and Tensor product

## Trivial group

In mathematics, a trivial group or zero group is a group consisting of a single element.

See Zero element and Trivial group

## Undergraduate Texts in Mathematics

Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag.

See Zero element and Undergraduate Texts in Mathematics

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

See Zero element and Union (set theory)

## Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that, or equivalently if the map from to that sends to is not injective. Zero element and zero divisor are 0 (number).

See Zero element and Zero divisor

## Zero morphism

In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Zero element and zero morphism are 0 (number).

See Zero element and Zero morphism

## Zero of a function

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) vanishes at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x). Zero element and zero of a function are 0 (number).

See Zero element and Zero of a function

## 0

0 (zero) is a number representing an empty quantity. Zero element and 0 are 0 (number).

## See also

### 0 (number)

- 0
- 0.0.0.0
- Additive identity
- Division by zero
- Empty sum
- Leading zero
- List of non-standard dates
- Names for the number 0
- Names for the number 0 in English
- Nilpotent
- Null (mathematics)
- Parity of zero
- Platform 0
- Root-finding algorithms
- Signed zero
- Slashed zero
- Symbols for zero
- Trailing zero
- Year zero
- Zero divisor
- Zero element
- Zero game
- Zero matrix
- Zero morphism
- Zero object (algebra)
- Zero of a function
- Zero ring
- Zero suppression
- Zero to the power of zero
- Zero-based numbering
- Zero-dimensional space
- Zero-energy universe
- Zero-product property

## References

Also known as 0 vector, 0-Free, List of zero terms, Zero element (disambiguation), Zero ideal, Zero tensor, Zero vector.