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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