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 ${\displaystyle {\mathcal {N}}_{R}}$ denote nilradical of a commutative ring ${\displaystyle R}$. There is a functor ${\displaystyle R\mapsto R/{\mathcal {N}}_{R}}$ of the category of commutative rings ${\displaystyle {\text{Crng}}}$ into the category of reduced rings ${\displaystyle {\text{Red}}}$ and it is left adjoint to the inclusion functor ${\displaystyle I}$ of ${\displaystyle {\text{Red}}}$ into ${\displaystyle {\text{Crng}}}$. The natural bijection ${\displaystyle {\text{Hom}}_{\text{Red}}(R/{\mathcal {N}}_{R},S)\cong {\text{Hom}}_{\text{Crng}}(R,I(S))}$ 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 ${\displaystyle {\mathfrak {p}}\mapsto \operatorname {dim} _{k({\mathfrak {p}})}(M\otimes k({\mathfrak {p}}))}$ 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]