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,
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- In the natural numbers, 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,
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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.
- 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 has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial.
- 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
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- 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. Thenproving that is trivial, i.e. The contrapositive, that if is non-trivial then 0 is not equal to 1, is therefore shown.