mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially in the field of ring theory, the term irreducible ring is used in a few different ways. * A (meet-)irreducible ring is a

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

in which the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of two non-

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

ideals is always non-zero. * A directly irreducible ring is a ring which cannot be written as the

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

of two non-

rings. * A subdirectly irreducible ring is a ring with a unique, non-zero minimum two-sided ideal. * A ring with an irreducible spectrum is a ring whose

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

is irreducible as a topological space. "Meet-irreducible" rings are referred to as "irreducible rings" in

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

. This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed. Meet-irreducible rings play an important part in commutative algebra, and directly irreducible and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irreducible algebras have also found use in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

. This article follows the convention that rings have

multiplicative identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

, but are not necessarily

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

Definitions

The terms "meet-reducible", "directly reducible" and "subdirectly reducible" are used when a ring is ''not'' meet-irreducible, or ''not'' directly irreducible, or ''not'' subdirectly irreducible, respectively. The following conditions are equivalent for a commutative ring ''R'': * ''R'' is meet-irreducible; * the zero ideal in ''R'' is irreducible, i.e. the intersection of two non-zero ideals of ''A'' always is non-zero. The following conditions are equivalent for a ring ''R'': * ''R'' is directly irreducible; * ''R'' has no

central idempotent In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring (mathematics), ring is an element such that . That is, the element is idempotent under the ring's multiplication. Mathematical induction, Inductively the ...

s except for 0 and 1. The following conditions are equivalent for a ring ''R'': * ''R'' is subdirectly irreducible; * when ''R'' is written as a subdirect product of rings, then one of the projections of ''R'' onto a ring in the subdirect product is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

; * The intersection of all non-zero ideals of ''R'' is non-zero. The following conditions are equivalent for a commutative ring ''R'': * the

of ''R'' is irreducible. * ''R'' possesses exactly one

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 prime ideals. De ...

(this

prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...

may be the zero ideal);

Examples and properties

If ''R'' is subdirectly irreducible or meet-irreducible, then it is also directly irreducible, but the

converses Chuck Taylor All-Stars or Converse All Stars (also referred to as "Converse", "Chuck Taylors", "Chucks", "Cons", "All Stars", and "Chucky Ts") are sneakers manufactured by American fashion brand Converse (lifestyle wear), Converse (a subsidiary ...

are not true. * All

integral domain In mathematics, 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 setting for studying divisibilit ...

s are meet-irreducible, but not all integral domains are subdirectly irreducible (e.g. Z). In fact, a commutative ring is a domain if and only if it is both meet-irreducible and reduced. * A commutative ring is a domain if and only if its spectrum is irreducible and it is reduced.The Stacks project
Tag 01J2
/ref> * The

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. ...

Z/4Z is a ring which has all three senses of irreducibility, but it is not a domain. Its only proper ideal is 2Z/4Z, which is maximal, hence prime. The ideal is also minimal. * The

direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

of two non-zero rings is never directly irreducible, and hence is never meet-irreducible or subdirectly irreducible. For example, in Z × Z the intersection of the non-zero ideals × Z and Z × is equal to the zero ideal × . * Commutative directly irreducible rings are

connected ring In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring ''A'' that satisfies one of the following equivalent conditions: * ''A'' possesses no non-trivial (that is, not equal to 1 or 0) idempotent elemen ...

s; that is, their only

idempotent element Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

s are 0 and 1.

Generalizations

Commutative meet-irreducible rings play an elementary role in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, where this concept is generalized to the concept of an irreducible scheme.

Notes

{{reflist Commutative algebra Ring theory