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Wikipedia / 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.

Additive identities

An additive identity is the identity element in an additive group or monoid. Some examples of additive identity include:

The zero vector under vector addition; in a normed vector space.
The zero function or zero map, under pointwise addition
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. Examples include:

The empty set, which is an absorbing element under Cartesian product
The zero function or zero map under pointwise multiplication

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 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

A zero morphism in a category is a generalised absorbing element under function composition.

. In the category of groups.

Least elements

A least element in a partially ordered set or lattice.

Zero module

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

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

In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. Some examples of zero matrices are

The set of m × n matrices with entries in a ring K is the additive identity in K.

(with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In a matrix ring 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

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

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.