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Reduced ring

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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 SpecR. Then R is reduced if and only if every finitely generated module of locally constant rank is projective.[2]

Examples and non-examples

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  • 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

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

See also

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Notes

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  1. ^ 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.)
  2. ^ Eisenbud 1995, Exercise 20.13.

References

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  • N. Bourbaki, Commutative Algebra, Hermann Paris 1972, Chap. II, § 2.7
  • N. Bourbaki, Algebra, Springer 1990, Chap. V, § 6.7
  • Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer-Verlag. ISBN 0-387-94268-8.