Identity Element

In Mathematics , an identity element (or '''neutral element''') is a special type of element of a Set with respect to a Binary Operation on that set. It leaves other elements unchanged when combined with them. This is used for Group s and Related Concepts .

The term ''identity element'' is often shortened to ''identity'' when there is no possibility of confusion; we do so in this article.

) be a set ''S'' with a binary operation --- on it (known as a Magma ). Then an element ''e'' of ''S'' is called a left identity if ''e'' --- ''a'' = ''a'' for all ''a'' in ''S'', and a '''right identity''' if ''a'' --- ''e'' = ''a'' for all ''a'' in ''S''. If ''e'' is both a left identity and a right identity, then it is called a '''two-sided identity''', or simply an '''identity'''.

An identity with respect to addition is called an additive identity and an identity with respect to multiplication is called a '''multiplicative identity'''. The distinction is used most often for sets that support both binary operations (such as with Ring s).

EXAMPLES

) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if ''l'' is a left identity and ''r'' is a right identity then ''l'' = ''l'' --- ''r'' = ''r''. In particular, there can never be more than one two-sided identity.

	CATEGORIES ABOUT IDENTITY ELEMENT

	abstract algebra

	algebra

	binary operations

	identity element

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Identity Element