Additive identity

In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

Elementary examples

• The additive identity familiar from elementary mathematics is zero, denoted 0. For example,
• :
• In the natural numbers (if 0 is included), the integers the rational numbers the real numbers and the complex numbers the additive identity is 0. This says that for a number belonging to any of these sets,
• :

Formal definition

Let be a group that is closed under the operation of addition, denoted +. An additive identity for , denoted , is an element in such that for any element in ,

Further examples

• In a group, the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof).
• A ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below).
• In the ring of -by- matrices over a ring , the additive identity is the zero matrix, denoted or , and is the -by- matrix whose entries consist entirely of the identity element 0 in . For example, in the 2×2 matrices over the integers the additive identity is
• :
• In the quaternions, 0 is the additive identity.
• In the ring of functions from , the function mapping every number to 0 is the additive identity.
• In the additive group of vectors in the origin or zero vector is the additive identity.

Properties

The additive identity is unique in a group

Let be a group and let and in both denote additive identities, so for any in ,

It then follows from the above that

The additive identity annihilates ring elements

In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any in , . This follows because:

The additive and multiplicative identities are different in a non-trivial ring

Let be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let be any element of . Then

proving that is trivial, i.e. The contrapositive, that if is non-trivial then 0 is not equal to 1, is therefore shown.

See also

• 0 (number)
• Additive inverse
• Identity element
• Multiplicative identity

References

Bibliography

• David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, .

External links