Editing Zero divisor (section)

{{Short description|Ring element that can be multiplied by a non-zero element to equal 0}}
{{distinguish|Division by zero}}
{{Use American English|date = March 2019}}
In [[abstract algebra]], an [[element (mathematics)|element]] {{math|''a''}} of a [[ring (algebra)|ring]] {{math|''R''}} is called a '''left zero divisor''' if there exists a nonzero {{math|''x''}} in {{math|''R''}} such that {{math|1=''ax'' = 0}},<ref>{{citation |author= N. Bourbaki |author-link= N. Bourbaki |title=Algebra I, Chapters 1–3 |page=98 |publisher=Springer-Verlag |year=1989}}</ref> or equivalently if the [[function (mathematics)|map]] from {{math|''R''}} to {{math|''R''}} that sends {{math|''x''}} to {{math|''ax''}} is not [[injective]].{{efn|1=Since the map is not injective, we have {{math|1=''ax'' = ''ay''}}, in which {{math|''x''}} differs from {{math|''y''}}, and thus {{math|1=''a''(''x'' − ''y'') = 0}}.}}  Similarly, an element {{math|''a''}} of a ring is called a '''right zero divisor''' if there exists a nonzero {{math|''y''}} in {{math|''R''}} such that {{math|1=''ya'' = 0}}.  This is a partial case of [[divisibility (ring theory)|divisibility in rings]].  An element that is a left or a right zero divisor is simply called a '''zero divisor'''.<ref>{{citation |author= Charles Lanski |year=2005 |title=Concepts in Abstract Algebra |publisher=American Mathematical Soc. |page=342 }}</ref> An element&nbsp;{{math|''a''}} that is both a left and a right zero divisor is called a '''two-sided zero divisor''' (the nonzero {{math|''x''}} such that {{math|1=''ax'' = 0}} may be different from the nonzero {{math|''y''}} such that {{math|1=''ya'' = 0}}). If the ring is [[commutative ring|commutative]], then the left and right zero divisors are the same.

An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called '''left regular''' or '''left cancellable''' (respectively, '''right regular''' or '''right cancellable''').
An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called '''regular''' or '''cancellable''',{{refn|{{cite book|author=Nicolas Bourbaki|year=1998|title=Algebra I|publisher=[[Springer Science+Business Media]]|page=15}}}} or a '''non-zero-divisor'''.  A zero divisor that is nonzero is called a '''nonzero zero divisor''' or a '''nontrivial zero divisor'''.  A non-[[zero ring|zero]] ring with no nontrivial zero divisors is called a [[domain (ring theory)|domain]].