Editing Zero divisor (section)

== Zero as a zero divisor ==
There is no need for a separate convention for the case {{math|1=''a'' = 0}}, because the definition applies also in this case: 
* If {{math|''R''}} is a ring other than the [[zero ring]], then {{math|0}} is a (two-sided) zero divisor, because any nonzero element {{mvar|x}} satisfies {{math|1=0''x'' = 0 = ''x''&hairsp;0}}.
* If {{math|''R''}} is the zero ring, in which {{math|0 {{=}} 1}}, then {{math|0}} is not a zero divisor, because there is no ''nonzero'' element that when multiplied by {{math|0}} yields {{math|0}}.

Some references include or exclude {{math|0}} as a zero divisor in ''all'' rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:
* In a commutative ring {{math|''R''}}, the set of non-zero-divisors is a [[multiplicative set]] in {{mvar|R}}.  (This, in turn, is important for the definition of the [[total quotient ring]].)  The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
* In a commutative [[noetherian ring]] {{math|''R''}}, the set of zero divisors is the [[union (set theory)|union]] of the [[associated prime|associated prime ideals]] of {{math|''R''}}.