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 x in the set, yields x. 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 edit

  • 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 n belonging to any of these sets,

Formal definition edit

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


Further examples edit

  • 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 Mm × n(R) of m-by-n matrices over a ring R, the additive identity is the zero matrix,[1] denoted O or 0, and is the m-by-n matrix whose entries consist entirely of the identity element 0 in R. 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 edit

The additive identity is unique in a group edit

Let (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G,


It then follows from the above that


The additive identity annihilates ring elements edit

In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any s in S, s · 0 = 0. This follows because:


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

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


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

See also edit

References edit

  1. ^ Weisstein, Eric W. "Additive Identity". Retrieved 2020-09-07.

Bibliography edit

  • David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, ISBN 0-471-43334-9.

External links edit

  • Uniqueness of additive identity in a ring at PlanetMath.