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

l{N}R

denote nilradical of a commutative ring

R

. There is a functor

R\mapstoR/l{N}R

of the category of commutative rings

Crng

into the category of reduced rings

Red

and it is left adjoint to the inclusion functor

I

of

Red

into

Crng

. 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)}(Mk(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

Generalizations

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

References

Notes and References

  1. Proof: let

    ak{p}i

    be all the (possibly zero) minimal prime ideals.

    D\subset\cupak{p}i:

    Let x be in D. Then xy = 0 for some nonzero y. Since R is reduced, (0) is the intersection of all

    ak{p}i

    and thus y is not in some

    ak{p}i

    . Since xy is in all

    ak{p}j

    ; in particular, in

    ak{p}i

    , x is in

    ak{p}i

    .

    D\supsetak{p}i:

    (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

    R\toR[S-1]

    . Let

    ak{q}

    be the pre-image of a maximal ideal. Then

    ak{q}

    is contained in both D and

    ak{p}

    and by minimality

    ak{q}=ak{p}

    . (This direction is immediate if R is Noetherian by the theory of associated primes.)