Reduced Ring

A reduced ring is a commutative ring with multiplicative identity that has no nonzero nilpotent elements. Equivalently, for some positive integer implies . Thus a commutative ring with multiplicative identity is reduced iff its nilradical is the zero ideal (Milne 2014, p. 5).

For an ideal of , the quotient ring is reduced iff equals its ideal radical . In particular, is always reduced. Every integral domain is reduced, but a reduced ring may have zero divisors. For example, the direct product of a field with itself is reduced, although . In contrast, the ring of dual numbers is not reduced because the class of is a nonzero nilpotent element.