Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0.
A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.
The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero.
Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero.
A quotient ring R/I is reduced if and only if I is a radical ideal.
denote nilradical of a commutative ring
and it is left adjoint to the inclusion functor
is induced from the universal property of quotient rings.
Let D be the set of all zero-divisors in a reduced ring R. Then D is the union of all minimal prime ideals.
[1] Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if
is a locally constant (or equivalently continuous) function on Spec R. Then R is reduced if and only if every finitely generated module of locally constant rank is projective.
[2] Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the notion of a reduced scheme.