Reduced ring explained
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 HomRed(R/l{N}R,S)\congHomCrng(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
ak{p}\mapsto\operatorname{dim}k(ak{p)}(M ⊗ k(ak{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.
References
- N. Bourbaki, Commutative Algebra, Hermann Paris 1972, Chap. II, § 2.7
- N. Bourbaki, Algebra, Springer 1990, Chap. V, § 6.7
- Book: Eisenbud, David . David Eisenbud . Commutative Algebra with a View Toward Algebraic Geometry . Graduate Texts in Mathematics . Springer-Verlag . 1995 . 0-387-94268-8.
Notes and References
- 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=\{xy|x\inR-D,y\inR-ak{p}\}
. 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.)