Definition
A reduced ring is a ring (typically assumed commutative with identity) that contains no non‑zero nilpotent elements; that is, if $a\in R$ satisfies $a^{n}=0$ for some positive integer $n$, then $a=0$. Equivalently, the nilradical of the ring (the set of all nilpotent elements) is the zero ideal.
Overview
Reduced rings form a fundamental class of commutative algebraic structures. Every integral domain is reduced, but the converse is false: a reduced ring may contain zero‑divisors. Classic examples include finite direct products of fields, Boolean rings (where every element is idempotent), and coordinate rings of reduced algebraic varieties. Reducedness is preserved under several standard constructions: subrings of a reduced ring are reduced, and quotients of a reduced ring by a radical ideal remain reduced. Conversely, a quotient of a reduced ring by an arbitrary ideal need not be reduced; the quotient is reduced precisely when the ideal is radical.
In algebraic geometry, the coordinate ring of a reduced scheme is a reduced ring; thus the term “reduced” reflects the geometric notion of a variety without embedded nilpotent structure.
Etymology/Origin
The adjective “reduced” is borrowed from the language of algebraic geometry, where a reduced scheme is one whose structure sheaf has no nilpotent sections. The concept of a reduced ring was formalized in the early 20th century within commutative algebra, building on the work of mathematicians such as Emmy Noether and Wolfgang Krull on radicals and prime ideals. The term emphasizes the removal (“reduction”) of nilpotent elements from a general ring.
Characteristics
- Nilradical: $\operatorname{Nil}(R)={a\in R\mid a^{n}=0\text{ for some }n\ge1}=0$.
- Prime intersection: The intersection of all prime ideals of $R$ is zero, i.e. $\bigcap_{\mathfrak p\in \operatorname{Spec}(R)}\mathfrak p =0$.
- Subdirect product representation: A commutative reduced ring is isomorphic to a subdirect product of integral domains; equivalently, it embeds into a direct product of fields.
- Zero‑divisors: Reduced rings may have zero‑divisors, but any zero‑divisor cannot be nilpotent.
- Localization: Localizing a reduced ring at any multiplicative set yields another reduced ring.
- Total ring of fractions: For a reduced ring $R$, the total ring of fractions $Q(R)$ (obtained by inverting all non‑zerodivisors) is a von Neumann regular ring; if $R$ is also Noetherian, $Q(R)$ is a finite product of fields.
Related Topics
- Nilpotent element, nilradical, Jacobson radical
- Integral domain, field, Boolean ring
- Radical ideal, prime ideal, spectrum of a ring ($\operatorname{Spec}(R)$)
- Reduced scheme, algebraic variety, coordinate ring
- Zero divisor, von Neumann regular ring, total ring of fractions
These concepts are interconnected in the study of commutative algebra and its applications to algebraic geometry.