Zero element
This article needs additional citations for verification. (August 2020) |
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
An additive identity is the identity element in an additive group or monoid. 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 whose components are all 0; in a normed vector spaceits norm (length) is also 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
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
A
- 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
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 0XY : 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 ∘ 0XY = 0XB and 0XY ∘ f = 0AY.
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 0XY : 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
A
Zero module
In
Zero ideal
In mathematics, the zero ideal in a ring is the ideal consisting of only the additive identity (or
Zero matrix
In
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 a matrix ring, the zero matrix serves the role of both an additive identity and an absorbing element. 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
Zero tensor
In
Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Among tensors of a given type, the zero tensor of that type serves as the additive identity among those tensors.
See also
- Null semigroup
- Zero divisor
- Zero object
- Zero of a function
- Zero— non-mathematical uses
References
- ISBN 978-981-13-0925-0.
- ISBN 9780387964126.
We have a zero matrix in which for all . ... We shall write it .