Editing Identity element (section)

==Definitions==
Let {{math|(''S'', ∗)}} be a set&nbsp;{{mvar|S}} equipped with a [[binary operation]]&nbsp;∗. Then an element&nbsp;{{mvar|e}} of&nbsp;{{mvar|S}} is called a {{visible anchor|left identity element|text='''[[left and right (algebra)|left]] identity'''}} if {{math|1=''e'' ∗ ''s'' = ''s''}} for all&nbsp;{{mvar|s}} in&nbsp;{{mvar|S}}, and a {{visible anchor|right identity element|text='''[[left and right (algebra)|right]] identity'''}} if {{math|1=''s'' ∗ ''e'' = ''s''}} for all&nbsp;{{mvar|s}} in&nbsp;{{mvar|S}}.<ref>{{harvtxt|Fraleigh|1976|p=21}}</ref> If {{mvar|e}} is both a left identity and a right identity, then it is called a '''{{visible anchor|two-sided identity}}''', or simply an '''{{visible anchor|identity}}'''.<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=96}}</ref><ref>{{harvtxt|Fraleigh|1976|p=18}}</ref><ref>{{harvtxt|Herstein|1964|p=26}}</ref><ref>{{harvtxt|McCoy|1973|p=17}}</ref><ref>{{Cite web|url=https://brilliant.org/wiki/identity-element/|title=Identity Element {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-01}}</ref>

An identity with respect to addition is called an [[Additive identity|{{visible anchor|additive identity}}]] (often denoted as&nbsp;0) and an identity with respect to multiplication is called a '''{{visible anchor|multiplicative identity}}''' (often denoted as&nbsp;1).<ref name=":0" /> These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a [[Group (mathematics)|group]] for example, the identity element is sometimes simply denoted by the symbol <math>e</math>. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as [[ring (mathematics)|ring]]s, [[integral domain]]s, and [[field (mathematics)|field]]s. The multiplicative identity is often called '''{{visible anchor|unity}}''' in the latter context (a ring with unity).<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=135}}</ref><ref>{{harvtxt|Fraleigh|1976|p=198}}</ref><ref>{{harvtxt|McCoy|1973|p=22}}</ref> This should not be confused with a [[unit (ring theory)|unit]] in ring theory, which is any element having a [[multiplicative inverse]]. By its own definition, unity itself is necessarily a unit.<ref>{{harvtxt|Fraleigh|1976|pp=198,266}}</ref><ref>{{harvtxt|Herstein|1964|p=106}}</ref>