# Zero element

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In mathematics, a **zero element** is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.

## Additive identities[edit]

An **additive identity** is the identity element in an additive group. It corresponds to the element 0 such that for all x in the group, 0 + *x* = *x* + 0 = *x*. Some examples of additive identity include:

- The
**zero vector**under vector addition: the vector of length 0 and whose components are all 0. Often denoted as or .^{[1]} - The
**zero function**or**zero map**defined by*z*(*x*) = 0, under pointwise addition (*f*+*g*)(*x*) =*f*(*x*) +*g*(*x*) - The
**empty set**under set union - An
**empty sum**or**empty coproduct** - An
**initial object**in a category (an empty coproduct, and so an identity under coproducts)

## Absorbing elements[edit]

An **absorbing element** in a multiplicative semigroup or semiring generalises the property 0 ⋅ *x* = 0. Examples include:

- The
**empty set**, which is an absorbing element under Cartesian product of sets, since { } ×*S*= { } - The
**zero function**or**zero map**defined by*z*(*x*) = 0 under pointwise multiplication (*f*⋅*g*)(*x*) =*f*(*x*) ⋅*g*(*x*)

Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a **field** or **ring**, which is both the additive identity and the multiplicative absorbing element, and whose principal ideal is the smallest ideal.

## Zero objects[edit]

A **zero object** in a category is both an initial and terminal object (and so an identity under both coproducts and products). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:

- The
**trivial group**, containing only the identity (a zero object in the category of groups) - The
**zero module**, containing only the identity (a zero object in the category of modules over a ring)

## Zero morphisms[edit]

A **zero morphism** in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0_{XY} : *X* → *Y* is the zero morphism among morphisms from *X* to *Y*, and *f* : *A* → *X* and *g* : *Y* → *B* are arbitrary morphisms, then *g* ∘ 0_{XY} = 0_{XB} and 0_{XY} ∘ *f* = 0_{AY}.

If a category has a zero object **0**, then there are canonical morphisms *X* → **0** and **0** → *Y*, and composing them gives a zero morphism 0_{XY} : *X* → *Y*. In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function *z*(*x*) = 0.

## Least elements[edit]

A **least element** in a partially ordered set or lattice may sometimes be called a zero element, and written either as 0 or ⊥.

## Zero module[edit]

In mathematics, the **zero module** is the module consisting of only the additive identity for the module's addition function. In the integers, this identity is zero, which gives the name *zero module*. That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.

## Zero ideal[edit]

In mathematics, the **zero ideal** in a ring is the ideal consisting of only the additive identity (or zero element). The fact that this is an ideal follows directly from the definition.

## Zero matrix[edit]

In mathematics, particularly linear algebra, a **zero matrix** is a matrix with all its entries being zero. It is alternately denoted by the symbol .^{[2]} Some examples of zero matrices are

The set of *m* × *n* matrices with entries in a ring *K* forms a module . The zero matrix in is the matrix with all entries equal to , where is the additive identity in *K*.

The zero matrix is the additive identity in . That is, for all :

There is exactly one zero matrix of any given size *m* × *n* (with entries from a given ring), so when the context is clear, one often refers to *the* zero matrix. In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix also represents the linear transformation which sends all vectors to the zero vector.

## Zero tensor[edit]

In mathematics, the **zero tensor** is a tensor, of any order, all of whose components are zero. The zero tensor of order 1 is sometimes known as the zero vector.

Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.

## See also[edit]

- Null semigroup
- Zero divisor
- Zero object
- Zero of a function
- Zero — non-mathematical uses

## References[edit]

**^**Nair, M. Thamban; Singh, Arindama (2018).*Linear Algebra*. Springer. p. 3. doi:10.1007/978-981-13-0926-7. ISBN 978-981-13-0925-0.**^**Lang, Serge (1987).*Linear Algebra*. Undergraduate Texts in Mathematics. Springer. p. 25. ISBN 9780387964126.We have a zero matrix in which for all . ... We shall write it .