Reduced ring

In ring theory, a ring R 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. In ring theory, a ring R 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 D be the set of all zerodivisors 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 p ↦ dim k ( p ) ⁡ ( M ⊗ k ( p ) ) {displaystyle {mathfrak {p}}mapsto operatorname {dim} _{k({mathfrak {p}})}(Motimes 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. Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of a reduced scheme.