Irreducible element

Last updated September 14, 2024

In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.

The irreducible elements are the terminal elements of a factorization process; that is, they are the factors that cannot be further factorized. The irreducible factors of an element are uniquely defined, up to the multiplication by a unit, if the integral domain is a unique factorization domain. It was discovered in the 19th century that the rings of integers of some number fields are not unique factorization domains, and, therefore, that some irreducible elements can appear in some factorization of an element and not in other factorizations of the same element. The ignorance of this fact is the main error in many of the wrong proofs of Fermat's Last Theorem that were given during the three centuries between Fermat's statement and Wiles's proof of Fermat's Last Theorem.

If $R$ is an integral domain, then $a$ is an irreducible element of $R$ if and only if, for all $b,c\in R$ , the equation $a=bc$ implies that the ideal generated by $a$ is equal to the ideal generated by $b$ or equal to the ideal generated by $c$ . This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no nonzero zero divisors is commonly made in the definition of irreducible elements. It results also that there are several ways to extend the definition of an irreducible element to an arbitrary commutative ring.^[1]

Relationship with prime elements

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element $a$ in a commutative ring $R$ is called prime if, whenever $a\mid bc$ for some $b$ and $c$ in $R,$ then $a\mid b$ or $a\mid c.$ ) In an integral domain, every prime element is irreducible,^{[lower-alpha 1]}^[2] but the converse is not true in general. The converse is true for unique factorization domains ^[2] (or, more generally, GCD domains).

Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if $D$ is a GCD domain and $x$ is an irreducible element of $D$ , then as noted above $x$ is prime, and so the ideal generated by $x$ is a prime (hence irreducible) ideal of $D$ .

Example

In the quadratic integer ring $\mathbf {Z} [{\sqrt {-5}}],$ it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,

3\mid \left(2+{\sqrt {-5}}\right)\left(2-{\sqrt {-5}}\right)=9,

but 3 does not divide either of the two factors.^[3]

Notes

↑ Consider $p$ a prime element of $R$ and suppose $p=ab.$ Then $p\mid ab,$ so $p\mid a$ or $p\mid b.$ Say $p\mid a,$ so $a=pc$ for some $c\in R$ . Then we have $p=ab=pcb,$ and so $p(1-cb)=0.$ Because $R$ is an integral domain we have $cb=1.$ Therefore $b$ is a unit and $p$ is irreducible.

Related Research Articles

In mathematics, more specifically in ring theory, a Euclidean domain is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them. Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.

<span class="mw-page-title-main">Fundamental theorem of arithmetic</span> Integers have unique prime factorizations

In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example,

In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers $x$ , $y$ , the greatest common divisor of $x$ and $y$ is denoted $. For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4 .$

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 divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

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 the zero ideal.

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal. Some authors such as Bourbaki refer to PIDs as principal rings.

In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.

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 prime elements from irreducible elements, a concept that is the same in UFDs but not the same in general.

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 specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In mathematics, specifically ring theory, a principal ideal is an ideal $in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in$

In mathematics, the ideal class group of an algebraic number field $K$ is the quotient group $J K / P K$ where $J K$ is the group of fractional ideals of the ring of integers of $K$ , and $P K$ is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of $K$ . The order of the group, which is finite, is called the class number of $K$ .

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 for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial $x 2 - 2$ is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as $if it is considered as a polynomial with real coefficients. One says that the polynomial x 2 - 2 is irreducible over the integers but not over the reals.$

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals. The theorem was first proven by Emanuel Lasker for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether.

In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization domain. Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials.

In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM).

In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is also a principal ideal. This means that Bézout's identity holds for every pair of elements, and that every finitely generated ideal is principal. Bézout domains are a form of Prüfer domain.

In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:

This is a glossary of commutative algebra.

References

↑ Anderson, D. D.; Valdes-Leon, Silvia (1996-06-01). "Factorization in Commutative Rings with Zero Divisors". Rocky Mountain Journal of Mathematics. 26 (2): 439–480. doi: 10.1216/rmjm/1181072068 . ISSN 0035-7596.
1 2 Sharpe, David (1987). Rings and factorization . Cambridge University Press. p. 54. ISBN 0-521-33718-6. Zbl 0674.13008.
↑ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[2] Consider $p$ a prime element of $R$ and suppose $p=ab.$ Then $p\mid ab,$ so $p\mid a$ or $p\mid b.$ Say $p\mid a,$ so $a=pc$ for some $c\in R$ . Then we have $p=ab=pcb,$ and so $p(1-cb)=0.$ Because $R$ is an integral domain we have $cb=1.$ Therefore $b$ is a unit and $p$ is irreducible.

[1] Anderson, D. D.; Valdes-Leon, Silvia (1996-06-01). "Factorization in Commutative Rings with Zero Divisors". Rocky Mountain Journal of Mathematics. 26 (2): 439–480. doi: 10.1216/rmjm/1181072068 . ISSN 0035-7596.

[Sha54-3] 1 2 Sharpe, David (1987). Rings and factorization . Cambridge University Press. p. 54. ISBN 0-521-33718-6. Zbl 0674.13008.

[4] William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9

[1]

[lower-alpha 1]

[2]

[3]