Ring without non-zero nilpotent elements
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
. There is a functor
of the category of commutative rings
into the category of reduced rings
and it is left adjoint to the inclusion functor
of
into
. 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.[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]
- ^ Proof: let
be all the (possibly zero) minimal prime ideals.
Let x be in D. Then xy = 0 for some nonzero y. Since R is reduced, (0) is the intersection of all
and thus y is not in some
. Since xy is in all
; in particular, in
, x is in
.
(stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript i. Let
. S is multiplicatively closed and so we can consider the localization
. Let
be the pre-image of a maximal ideal. Then
is contained in both D and
and by minimality
. (This direction is immediate if R is Noetherian by the theory of associated primes.)
- ^ Eisenbud 1995, Exercise 20.13.