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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 (often denoted as 0) and an identity with respect to multiplication is called a '''multiplicative identity''' (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as Ring s. The multiplicative identity is often called the '''unit''' in the latter context, where, unfortunately, a Unit is also sometimes used to mean an element with a multiplicative inverse.


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. If there were two, ''e'' and ''f'', then ''e'' --- ''f'' would have to be equal to both ''e'' and ''f''.



SEE ALSO