In
algebra
Algebra () is one of the 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 mathematics.
Elementary a ...
, an irreducible element of a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
is a non-zero element that is not
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
(that is, is not a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
), and is not the product of two non-invertible elements.
Relationship with prime elements
Irreducible elements should not be confused with
prime element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish pri ...
s. (A non-zero non-unit element
in a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
is called prime if, whenever
for some
and
in
then
or
) In an
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 ...
, every prime element is irreducible,
[Sharpe (1987) p.54] but the converse is not true in general. The converse is true for
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
s
[ (or, more generally, GCD domains).
Moreover, while an ideal generated by a prime element is a ]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 ...
, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals..
Examples
* Every prime ideal is irreducible. Let J and K be ideals of a commutative ring ...
. However, if is a GCD domain and is an irreducible element of , then as noted above is prime, and so the ideal generated by is a prime (hence irreducible) ideal of .
Example
In the quadratic integer ring
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
:
with and (usual) integers. When algebrai ...
it can be shown using norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,
:
but 3 does not divide either of the two factors.[William W. Adams and Larry Joel Goldstein (1976), ''Introduction to Number Theory'', p. 250, Prentice-Hall, Inc., ]
See also
* Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
References
*
{{DEFAULTSORT:Irreducible Element
Ring theory
Algebraic properties of elements