Editing Reduced ring (section)

==Examples and non-examples==
* [[Subring]]s, [[product of rings|products]], and [[localization of a ring|localizations]] of reduced rings are again reduced rings.
* The ring of [[Integer#Algebraic properties|integers]] '''Z''' is a reduced ring. Every [[Field (mathematics)|field]] and every [[polynomial ring]] over a field (in arbitrarily many variables) is a reduced ring.
* More generally, every [[integral domain]] is a reduced ring since a nilpotent element is a fortiori a [[zero-divisor]]. On the other hand, not every reduced ring is an integral domain; for example, the ring '''Z'''[''x'',&thinsp;''y'']/(''xy'') contains ''x'' + (''xy'') and ''y'' + (''xy'') as zero-divisors, but no non-zero nilpotent elements. As another example, the ring '''Z'''&thinsp;×&thinsp;'''Z''' contains (1,&nbsp;0) and (0,&thinsp;1) as zero-divisors, but contains no non-zero nilpotent elements.
* The ring '''Z'''/6'''Z''' is reduced, however '''Z'''/4'''Z''' is not reduced: the class 2&nbsp;+&nbsp;4'''Z''' is nilpotent.  In general, '''Z'''/''n'''''Z''' is reduced if and only if ''n''&nbsp;=&nbsp;0 or ''n'' is [[square-free integer|square-free]].
* If ''R'' is a commutative ring and ''N'' is its [[nilradical of a ring|nilradical]], then the quotient ring ''R''/''N'' is reduced.
* A commutative ring ''R'' of [[prime number|prime]] [[characteristic (algebra)|characteristic]] ''p'' is reduced if and only if its [[Frobenius endomorphism]] is [[injective]] (cf. [[Perfect field]].)