Reduced ring

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 A form an ideal of A, the so-called nilradical of A; therefore a commutative ring is reduced if and only if its nilradical is reduced to zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero.

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

Generalizations

Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the concept 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