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Concept
Zero element
Summary
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.
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 .
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 (an empty coproduct, and so an identity under coproducts)
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.
A zero object in a is both an initial and terminal object (and so an identity under both coproducts and ). 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 )
The zero module, containing only the identity (a zero object in the category of modules over a ring)
A zero morphism in a is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism.
Official source
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