Editing Zero divisor (section)

== Properties ==
* In the ring of {{mvar|n}}&thinsp;×&thinsp;{{mvar|n}} matrices over a field, the left and right zero divisors coincide; they are precisely the [[singular matrix|singular matrices]]. In the ring of {{mvar|n}}&thinsp;×&thinsp;{{mvar|n}} matrices over an [[integral domain]], the zero divisors are precisely the matrices with [[determinant]] zero.
* Left or right zero divisors can never be [[unit (ring theory)|unit]]s, because if {{math|''a''}} is invertible and {{math|1=''ax'' = 0}} for some nonzero {{math|''x''}}, then {{math|1=0 = ''a''<sup>−1</sup>0 = ''a''<sup>−1</sup>''ax'' = ''x''}}, a contradiction.
* An element is [[Cancellation property|cancellable]] on the side on which it is regular.  That is, if {{math|''a''}} is a left regular, {{math|1=''ax'' = ''ay''}} implies that {{math|1=''x'' = ''y''}}, and similarly for right regular.