Identity element

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In mathematics, the identity or neutral element element is a special element type of a set with respect to binary operations in the set, which makes other elements unchanged when combined with them. This concept is used in algebraic structures such as groups and rings. The term identity element is often abbreviated to identity (as will be done in this article) when there is no possibility of confusion, but identity is implicitly dependent on the binary operation it is associated with.

Let ( S , *) to a set of S with binary * operations on it. Then an e element of S is called left identity if e * a = a for all a in S , and the correct identity if < i> a * a a for all a in S . If e is the correct identity and left identity, then it's called two-sided identity , or just an identity .

Identity with respect to additions is called an additional identity (often denoted as 0) and multiplicative identity is referred to as multiplication identity (often denoted as 1). This does not need the usual addition and multiplication, but rather an arbitrary operation. This difference is most commonly used for sets that support both binary operations, such as rings and fields. Multiplicative identity is often called unity in the last context (ring with unity). This should not be confused with the unit in ring theory, which is an element that has multiplication inverse. Unity itself is a unit.

Video Identity element

Example

Maps Identity element

Properties

As the last example (semigroup) shows, it is possible to ( S , *) have some left identity. In fact, every element can be a left identity. Similarly, there can be some true identity. But if there is a true identity and a left identity, then they are the same and there is only one double-sided identity. To see this, note that if l is the left identity and r is the correct identity then l = l * r = r . In particular, there will never be more than one two-sided identity. If there are two, e and f , then e * f will have that equal to e and f .

It is also possible for ( S , *) to have the no identity element. A common example of this is a cross-vector product; in this case, the absence of an identity element is due to the fact that the direction of any non-zero cross product is always orthogonal to every multiplied element - making it impossible to obtain non-zero vectors in the same direction as the original. Another example is an additional semigroup of positive original numbers.

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See also

• Absorbs the
element
• Invalid additive
• The inversion element
• Monoid
• Pseudo-ring
• Quasigroup
• Unital (disambiguation)

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References

• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Stories and Categories with Applications for Wreath Products and Graphs , De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p.Ã, 14-15

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