Английская Википедия: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 Шаблон:Mvar in the set, yields Шаблон:Mvar. 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,
- <math>5+0 = 5 = 0+5.</math>
- In the natural numbers Шаблон:Tmath (if 0 is included), the integers Шаблон:Tmath the rational numbers Шаблон:Tmath the real numbers Шаблон:Tmath and the complex numbers Шаблон:Tmath the additive identity is 0. This says that for a number Шаблон:Mvar belonging to any of these sets,
- <math>n+0 = n = 0+n.</math>
Formal definition
Let Шаблон:Mvar be a group that is closed under the operation of addition, denoted +. An additive identity for Шаблон:Mvar, denoted Шаблон:Mvar, is an element in Шаблон:Mvar such that for any element Шаблон:Mvar in Шаблон:Mvar,
- <math>e+n = n = n+e.</math>
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 Шаблон:Math of Шаблон:Mvar-by-Шаблон:Mvar matrices over a ring Шаблон:Mvar, the additive identity is the zero matrix,[1] denoted Шаблон:Math or Шаблон:Math, and is the Шаблон:Mvar-by-Шаблон:Mvar matrix whose entries consist entirely of the identity element 0 in Шаблон:Mvar. For example, in the 2×2 matrices over the integers Шаблон:Tmath the additive identity is
- <math>0 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}</math>
- In the quaternions, 0 is the additive identity.
- In the ring of functions from Шаблон:Tmath, the function mapping every number to 0 is the additive identity.
- In the additive group of vectors in Шаблон:Tmath the origin or zero vector is the additive identity.
Properties
The additive identity is unique in a group
Let Шаблон:Math be a group and let Шаблон:Math and Шаблон:Math in Шаблон:Mvar both denote additive identities, so for any Шаблон:Mvar in Шаблон:Mvar,
- <math>0+g = g = g+0, \qquad 0'+g = g = g+0'.</math>
It then follows from the above that
- <math>{\color{green}0'} = {\color{green}0'} + 0 = 0' + {\color{red}0} = {\color{red}0}.</math>
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 Шаблон:Mvar in Шаблон:Mvar, Шаблон:Math. This follows because:
- <math>\begin{align}
s \cdot 0 &= s \cdot (0 + 0) = s \cdot 0 + s \cdot 0 \\ \Rightarrow s \cdot 0 &= s \cdot 0 - s \cdot 0 \\ \Rightarrow s \cdot 0 &= 0.
\end{align}</math>
The additive and multiplicative identities are different in a non-trivial ring
Let Шаблон:Mvar be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let Шаблон:Mvar be any element of Шаблон:Mvar. Then
- <math>r = r \times 1 = r \times 0 = 0</math>
proving that Шаблон:Mvar is trivial, i.e. Шаблон:Math The contrapositive, that if Шаблон:Mvar is non-trivial then 0 is not equal to 1, is therefore shown.
See also
References
Bibliography
- David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, Шаблон:ISBN.
External links
