ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

, a branch of mathematics, a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

is called a reduced ring if it has no non-zero

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''² = 0 implies ''x'' = 0. A commutative

algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

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 Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

of ''R'', called the nilradical of ''R''; therefore a commutative ring is reduced

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

its nilradical is

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A

quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...

''R/I'' is reduced if and only if ''I'' is a

radical ideal In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is call ...

. Let ''D'' be the set of all

zero-divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...

s in a reduced ring ''R''. Then ''D'' is the union of all

minimal prime ideal In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Defini ...

s. Over a

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...

''R'', we say a finitely generated

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

''M'' has locally constant rank if

\mathfrak \mapsto \operatorname_(M \otimes k(\mathfrak))

is a locally constant (or equivalently continuous) function on Spec ''R''. Then ''R'' is reduced if and only if every finitely generated module of locally constant rank is projective.

Examples and non-examples

* Subrings,

products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

, and localizations of reduced rings are again reduced rings. * The ring of

integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

Z is a reduced ring. Every

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

and every

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...

over a field (in arbitrarily many variables) is a reduced ring. * More generally, every

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...

is a reduced ring since a nilpotent element is a fortiori a

. 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/''n''Z is reduced if and only if ''n'' = 0 or ''n'' is a

square-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square- ...

. * 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 A characteristic is a distinguishing feature of a person or thing. It may refer to: Computing * Characteristic (biased exponent), an ambiguous term formerly used by some authors to specify some type of exponent of a floating point number * Charact ...

''p'' for some

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

''p'' is reduced if and only if its

Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

(cf.

Perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k' ...

Generalizations

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

reduced scheme This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

Notes

References

N. Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...

, ''Commutative Algebra'', Hermann Paris 1972, Chap. II, § 2.7 *

, ''Algebra'', Springer 1990, Chap. V, § 6.7 * Eisenbud, David, ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, {{isbn, 0-387-94268-8. Ring theory pl:Element nilpotentny#Pierścień zredukowany