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