Zero Element - Absorbing Elements

Absorbing Elements

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 function multiplication, (f × g)(x) = f(x) × g(x), since z × f = z.

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.