Irreducible Ring
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the field of
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 re ...
, the term irreducible ring is used in a few different ways. * A (meet-)irreducible ring is 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 ...
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 i ...
of two non-
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
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. To see how the direct sum is used in abstract algebra, consider a more ...
of two non-
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
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 (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
is
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
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 ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
. 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 (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
. This article follows the convention that rings have
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
, 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. Most familiar as the name o ...
.


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 In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
, 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 abstract algebra, an idempotent element or simply idempotent of a ring is an element ''a'' such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for ...
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 In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however ne ...
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 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 them. The word is ...
; * The intersection of all non-zero ideals of ''R'' is non-zero. The following conditions are equivalent for a commutative ring ''R'': * the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of ''R'' is
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
. * ''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 primes. Definition ...
(this
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
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") is a model of casual shoe manufactured by Converse (a subsidiary of Nike, Inc. since 2003) that was i ...
are not true. * All
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 set ...
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, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
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 Ã—  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 elem ...
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, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, where this concept is generalized to the concept of an
irreducible scheme In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
.


Notes

{{reflist Commutative algebra Ring theory