In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. 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. Let denote nilradical of a commutative ring R. There is a functor of the category of commutative rings \text{Crng} into the category of reduced rings \text{Red} and it is left adjoint to the inclusion functor I of \text{Red} into \text{Crng}. The natural bijection 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. 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.

Examples and non-examples

Generalizations

Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the notion of a reduced scheme.