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※
- 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※
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※
- 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/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, 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※
The additive identity is unique in a group※
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※
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※
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※
References※
- ^ Weisstein, Eric W. "Additive Identity". mathworld.wolfram.com. Retrieved 2020-09-07.
Bibliography※
- David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, ISBN 0-471-43334-9.