Reduced ring

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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.[1]

Examples and non-examples[edit]

  • 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 a square-free integer.
  • If R is a commutative ring and N is the nilradical of R, then the quotient ring R/N is reduced.
  • A commutative ring R of characteristic p for some prime number p is reduced if and only if its Frobenius endomorphism is injective. (cf. perfect field.)

Generalizations[edit]

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

See also[edit]

Notes[edit]

  1. ^ Proof: let \mathfrak{p}_i be all the (possibly zero) minimal prime ideals.
    D \subset \cup \mathfrak{p}_i: Let x be in D. Then xy = 0 for some nonzero y. Since R is reduced, (0) is the intersection of all \mathfrak{p}_i and thus y is not in some \mathfrak{p}_i. Since xy is in all \mathfrak{p}_j; in particular, in \mathfrak{p}_i, x is in \mathfrak{p}_i.
    D \supset \mathfrak{p}_i: (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript i. Let S = \{ xy | x \in R - D, y \in R - \mathfrak{p} \}. S is multiplicatively closed and so we can consider the localization R \to R[S^{-1}]. Let \mathfrak{q} be the pre-image of a maximal ideal. Then \mathfrak{q} is contained in both D and \mathfrak{p} and by minimality \mathfrak{q} = \mathfrak{p}. (This direction is immediate if R is Noetherian by the theory of associated primes.)

References[edit]

  • N. Bourbaki, Commutative Algebra, Hermann Paris 1972, Chap. II, § 2.7
  • N. Bourbaki, Algebra, Springer 1990, Chap. V, § 6.7