Reduced ring

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In ring theory, a commutative ring R is called a reduced ring if it has no non-zero nilpotent elements. 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 A form an ideal of A, the so-called nilradical or nilpotent radical of A; therefore a commutative ring is reduced if and only if its nilpotent radical is reduced to zero. Each one of the following two statements is also equivalent to a commutative ring A being reduced:

• there exists an integer n>1 such that the map from A to A raising elements to the n-th power is injective;
• for any integer n>1 the map from A to A raising elements to the n-th power is injective.

[edit] Examples

• If A is an arbitrary commutative ring and N is the nilpotent radical of A then the quotient ring A/N is reduced.
• 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 non-zero reduced ring is an integral domain. Counterexamples: The ring Z[x, y]/(xy) contains x and y as zero divisors, but no non-zero nilpotent elements; the ring Z×Z contains (1,0) and (0,1) as zero divisors, but no non-zero nilpotent elements.

[edit] Generalizations

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

[edit] References

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