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Prime element
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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.
Definition
[edit]An element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever p divides ab for all a and b in R, then p divides a or p divides b. With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element p is prime if, and only if, the principal ideal (p) generated by p is a nonzero prime ideal.[1] (Note that in an integral domain, the ideal (0) is a prime ideal, but 0 is an exception in the definition of 'prime element'.)
Interest in prime elements comes from the fundamental theorem of arithmetic, which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers. This led to the study of unique factorization domains, which generalize what was just illustrated in the integers.
Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in Z but it is not in Z[i], the ring of Gaussian integers, since 2 = (1 + i)(1 − i) and 2 does not divide any factor on the right.
Connection with prime ideals
[edit]An ideal I in the ring R (with unity) is prime if the factor ring R/I is an integral domain. Equivalently, I is prime if whenever then either or .
In an integral domain, a nonzero principal ideal is prime if and only if it is generated by a prime element.
Irreducible elements
[edit]Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible[2] but the converse is not true in general. However, in unique factorization domains,[3] or more generally in GCD domains, primes and irreducibles are the same.
Examples
[edit]The following are examples of prime elements in rings:
- The integers ±2, ±3, ±5, ±7, ±11, ... in the ring of integers Z
- the complex numbers (1 + i), 19, and (2 + 3i) in the ring of Gaussian integers Z[i]
- the polynomials x2 − 2 and x2 + 1 in Z[x], the ring of polynomials over Z.
- 2 in the quotient ring Z/6Z
- x2 + (x2 + x) is prime but not irreducible in the ring Q[x]/(x2 + x)
- In the ring Z2 of pairs of integers, (1, 0) is prime but not irreducible (one has (1, 0)2 = (1, 0)).
- In the ring of algebraic integers the element 3 is irreducible but not prime (as 3 divides and 3 does not divide any factor on the right).
![{\displaystyle \mathbf {Z} [{\sqrt {-5}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ed4e4e8f7026e55edc2e4c5f68e4fc9c93da46)
References
[edit]- Notes
- ^ Hungerford 1980, Theorem III.3.4(i), as indicated in the remark below the theorem and the proof, the result holds in full generality.
- ^ Hungerford 1980, Theorem III.3.4(iii)
- ^ Hungerford 1980, Remark after Definition III.3.5
- Sources
- Section III.3 of Hungerford, Thomas W. (1980), Algebra, Graduate Texts in Mathematics, vol. 73 (Reprint of 1974 ed.), New York: Springer-Verlag, ISBN 978-0-387-90518-1, MR 0600654
- Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686, ISBN 0-7167-1933-9, MR 1009787
- Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021
Prime element
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In ring theory, a prime element of an integral domain is defined as a nonzero non-unit element such that if divides the product for any , then divides or divides .[1] This property mirrors the classical notion of primality in the integers, where prime numbers divide products only by dividing one factor.[2] Equivalently, is prime if and only if the principal ideal is a nonzero prime ideal in .[3]
Prime elements play a central role in the study of factorization in commutative rings, particularly in ensuring unique factorization up to units. In any integral domain, every prime element is irreducible, meaning it cannot be expressed as a product of two non-unit elements.[4] However, the converse does not always hold; there exist integral domains where irreducible elements are not prime, leading to non-unique factorizations.[5] In principal ideal domains (PIDs) and unique factorization domains (UFDs), such as the ring of integers or polynomial rings where is a field like , every irreducible element is prime, guaranteeing that every nonzero non-unit element factors uniquely into primes.[6]
The concept extends beyond basic integral domains to more advanced structures, including orders in algebraic number fields, where prime elements facilitate the decomposition of ideals and elements into products that reflect arithmetic properties.[7] Prime elements are foundational for theorems on divisibility, such as the fact that in Euclidean domains, factorization into primes is unique, underscoring their importance in algebraic number theory and commutative algebra.[8]
Definitions and Properties
Formal Definition
An integral domain is a commutative ring with unity that has no zero divisors. An element is called a unit if there exists another element such that . In such a ring, the divisibility relation is defined as follows: an element divides an element , denoted , if there exists some such that . A nonzero non-unit element is called prime if, whenever for some , then either or . This definition is distinct from that of an irreducible element, though the two concepts are related in certain rings.[9]Basic Properties
In an integral domain , a prime element is a nonzero non-unit such that whenever divides a product (with ), then divides or divides . This property ensures that prime elements behave analogously to prime numbers in the integers, capturing a form of divisibility that prevents "skipping" factors in products. A fundamental property is that every prime element is irreducible. To see this, suppose for some . Then divides , so by the prime property, divides or divides . Without loss of generality, assume divides , so for some . Substituting gives , and since is an integral domain, canceling yields , meaning is a unit. Thus, cannot be expressed as a product of two non-units, confirming irreducibility.[10][11] Prime elements also generate principal ideals that are prime ideals. Specifically, the ideal is a prime ideal if is prime. To verify, note that since is not a unit. Now suppose , so divides . By the prime property, divides or divides , hence or . This satisfies the definition of a prime ideal: a proper ideal where if the product of two elements is in the ideal, then at least one is in the ideal.[10][12] Conversely, in integral domains, a nonzero principal prime ideal is generated by a prime element. In integral domains, prime elements are non-zero-divisors. Suppose for some . Since is an integral domain (as is prime), the zero element in the quotient has no nonzero annihilators, implying . But and for some would imply , so . Since is a domain, , hence . These properties underpin factorization in integral domains. When primes and irreducibles coincide—meaning every irreducible element is prime—the domain admits unique factorization into irreducibles up to units and ordering, as seen in unique factorization domains, though full characterization requires additional conditions such as every irreducible generating a prime ideal.[11][12]Relations to Other Algebraic Structures
Connection to Irreducible Elements
In an integral domain, an irreducible element is defined as a non-zero, non-unit element that cannot be factored into the product of two non-unit elements. This contrasts with prime elements, which are non-zero, non-unit elements such that if the prime divides a product, it divides at least one of the factors. A fundamental relationship holds in integral domains: every prime element is irreducible.[13] The proof of this implication, which relies on the prime's divisibility property to show that any factorization must involve a unit, is detailed in the section on basic properties. The converse does not always hold; for instance, in the quadratic integer ring , the element 3 is irreducible but not prime, as it divides the product without dividing either factor.[14] Prime and irreducible elements coincide under specific conditions on the domain. In principal ideal domains (PIDs), where every ideal is generated by a single element, every irreducible element generates a prime ideal and thus is prime.[15] Similarly, in unique factorization domains (UFDs), where every non-zero non-unit element factors uniquely into irreducibles up to units and ordering, irreducibles are prime.[16] Historically, the integers exemplify a domain where primes and irreducibles coincide, providing early intuition through Euclid's Elements (circa 300 BCE), which established key results on prime factorization and the infinitude of primes.[17]Connection to Prime Ideals
In a commutative ring with identity, an ideal is a prime ideal if and whenever for , then or ; equivalently, the quotient ring is an integral domain.[18][19] A fundamental connection between prime elements and prime ideals arises in integral domains. Specifically, in an integral domain , a nonzero nonunit element is prime if and only if the principal ideal it generates is a nonzero prime ideal. To see this, suppose is prime and , so ; by the definition of primality, or , hence or , showing is prime. The converse follows similarly, as membership in corresponds to divisibility by .[19] However, not all prime ideals are principal. For instance, in the polynomial ring , the ideal is prime because the quotient , which is an integral domain (in fact, a field), but is not principal, as any single generator would need to divide both 2 and , which is impossible in .[20][21] In more structured rings like Dedekind domains, every nonzero prime ideal is maximal and thus has height one, reflecting the Krull dimension of one for such domains. Prime elements in a Dedekind domain generate principal prime ideals, which are a subset of these height-one primes; the domain is a unique factorization domain if and only if every height-one prime ideal is principal, generated by a prime element.[22][23]Examples and Applications
In Integral Domains
In the ring of integers , which is an integral domain, the prime elements are the associates of the positive prime numbers, such as , , and .[24] These elements satisfy the defining property: if a prime divides a product in , then divides or divides , as established by Euclid's lemma.[25] The units in are solely , ensuring that these primes are non-units and nonzero.[24] The Fundamental Theorem of Arithmetic asserts that every nonzero non-unit element in can be expressed uniquely as a product of these prime elements, up to ordering and association by units.[26] This unique factorization domain (UFD) property underscores the foundational role of prime elements in . Since is a principal ideal domain (PID), its prime elements coincide precisely with the irreducible elements.[27] Extending to the Gaussian integers , another integral domain, the element is a prime element.[28] Here, the ordinary prime 2 from ramifies, factoring as up to units, illustrating how prime elements behave in quadratic extensions.[28] As is itself a Euclidean domain and thus a PID, its prime elements also align with irreducibles.[28]In Polynomial Rings
In polynomial rings over a field , denoted , the ring is a principal ideal domain (PID), so every irreducible element is prime.[29] For example, the polynomial is irreducible over and thus prime in , as it generates a maximal ideal and , a field.[29] A key criterion for irreducibility (and hence primality) in is Eisenstein's criterion: if there exists a prime such that divides all coefficients except the leading one, and does not divide the constant term, then the polynomial is irreducible over .[30] For instance, is irreducible over for prime , as divides the lower coefficients but does not divide .[30] This often establishes primality directly in the UFD . In the polynomial ring , which is not a PID, prime elements include certain constants and non-constant polynomials. The content of a polynomial is the gcd of its coefficients; a primitive polynomial has content 1. Gauss's lemma states that the product of two primitive polynomials is primitive, and a primitive polynomial is irreducible in if and only if it is irreducible in .[31] Thus, irreducible primitive polynomials, like , are prime in . Additionally, the element is prime, as it divides every polynomial with zero constant term, and , an integral domain.[32] Constant primes from , such as 2, are also prime in , since , an integral domain.[32] In multivariable polynomial rings like over a field , identifying prime elements is more complex, as the ring is a UFD but not a PID for ; prime elements are irreducible polynomials that generate height-1 prime ideals, though non-principal prime ideals of higher height exist.[33]References
- https://commalg.subwiki.org/wiki/Irreducible_element_not_implies_prime