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.
Some examples of additive identity 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.
In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function z(x) = 0.
That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.
consisting of only the additive identity (or zero element).
The fact that this is an ideal follows directly from the definition.
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.