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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.
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.
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.
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.
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.
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.