Reduced ring

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 \mathfrak{N}_R denote the nilradical of a commutative ring R. There is a functor R \mapsto R/\mathfrak{N}_R of the category of commutative rings \textsf{CRing} into the category of reduced rings \textsf{Red} and it is left adjoint to the inclusion functor I of \textsf{Red} into \textsf{CRing}, making \textsf{Red} a reflective subcategory of \textsf{CRing}. The natural bijection \text{Hom}_{\textsf{Red}}(R/\mathfrak{N}_R,S)\cong\text{Hom}_{\textsf{CRing}}(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. Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if \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. ==Examples and non-examples==

• Subrings, products, and localizations of reduced rings are again reduced rings. • The ring of integers Z is a reduced ring. Every 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, y]/(xy) contains x + (xy) and y + (xy) as zero-divisors, but no non-zero nilpotent elements. As another example, the ring Z × Z contains (1, 0) and (0, 1) as zero-divisors, but contains no non-zero nilpotent elements. • The ring Z/6Z is reduced, however Z/4Z is not reduced: the class 2 + 4Z is nilpotent. In general, Z/nZ is reduced if and only if n = 0 or n is square-free. • If R is a commutative ring and N is its nilradical, then the quotient ring R/N is reduced. • A commutative ring R of prime characteristic p is reduced if and only if its Frobenius endomorphism is injective (cf. Perfect field.) ==Generalizations==

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