Identity element
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==Definitions== |
==Definitions== |
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Let {{ |
Let {{nul|(''S'', ∗)}} be a set {{mvar|S}} equipped with a [[binary operation]] ∗. Then an element {{mvar|e}} of {{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 {{mvar|s}} in {{mvar|S}}, and a {{visible anchor|right identity element|text='''[[left and right (algebra)|right]] identity'''}} if {{math|1=''s'' ∗ ''e'' = ''s''}} for all {{mvar|s}} in {{mvar|S}}.{{harvtxt|Fraleigh|1976|p=21}} 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}}'''.{{harvtxt|Beauregard|Fraleigh|1973|p=96}}{{harvtxt|Fraleigh|1976|p=18}}{{harvtxt|Herstein|1964|p=26}}{{harvtxt|McCoy|1973|p=17}}{{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}} |
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An identity with respect to addition is called an [[Additive identity|{{visible anchor|additive identity}}]] (often denoted as 0) and an identity with respect to multiplication is called a '''{{visible anchor|multiplicative identity}}''' (often denoted as 1). 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 . 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).{{harvtxt|Beauregard|Fraleigh|1973|p=135}}{{harvtxt|Fraleigh|1976|p=198}}{{harvtxt|McCoy|1973|p=22}} 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.{{harvtxt|Fraleigh|1976|pp=198,266}}{{harvtxt|Herstein|1964|p=106}} |
An identity with respect to addition is called an [[Additive identity|{{visible anchor|additive identity}}]] (often denoted as 0) and an identity with respect to multiplication is called a '''{{visible anchor|multiplicative identity}}''' (often denoted as 1). 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 . 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).{{harvtxt|Beauregard|Fraleigh|1973|p=135}}{{harvtxt|Fraleigh|1976|p=198}}{{harvtxt|McCoy|1973|p=22}} 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.{{harvtxt|Fraleigh|1976|pp=198,266}}{{harvtxt|Herstein|1964|p=106}} |
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