# Identity element

An identity element in a set $$S$$ with a binary operation $$*$$ is an element $$i$$ that leaves any element $$a \in S$$ unchanged when combined with it in that operation.

Formally, we can define an element $$i$$ to be an identity element if the following two statements are true:

1. For all $$a \in S$$, $$i \* a = a$$. If only this statement is true then $$i$$ is said to be a left identity.

2. For all $$a \in S$$, $$a \* i = a$$. If only this statement is true then $$i$$ is said to be a right identity.

The existence of an identity element is a property of many algebraic structures, such as groups, rings, and fields.

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