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Prime element

In ring theory, a prime element of an integral domain R is defined as a nonzero non-unit element p \in R such that if p divides the product ab for any a, b \in R, then p divides a or p divides b.^[1] This property mirrors the classical notion of primality in the integers, where prime numbers divide products only by dividing one factor.^[2] Equivalently, p is prime if and only if the principal ideal (p) is a nonzero prime ideal in R.^[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 \mathbb{Z} or polynomial rings k where k is a field like \mathbb{Q}, 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 u \in R is called a unit if there exists another element v \in R such that u v = v u = 1. In such a ring, the divisibility relation is defined as follows: an element a \in R divides an element b \in R, denoted a \mid b, if there exists some c \in R such that b = a c. A nonzero non-unit element p \in R is called prime if, whenever p \mid a b for some a, b \in [R](/page/R), then either p \mid a or p \mid b. 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 R, a prime element p \in R is a nonzero non-unit such that whenever p divides a product ab (with a, b \in R), then p divides a or p divides b. 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 p = ab for some a, b \in R. Then p divides ab = p, so by the prime property, p divides a or p divides b. Without loss of generality, assume p divides a, so a = pc for some c \in R. Substituting gives p = pcb, and since R is an integral domain, canceling p yields $1 = cb, meaning b is a unit. Thus, p 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 (p) = \{ pr \mid r \in R \} is a prime ideal if p is prime. To verify, note that (p) \neq R since p is not a unit. Now suppose ab \in (p), so p divides ab. By the prime property, p divides a or p divides b, hence a \in (p) or b \in (p). 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 p b = 0 for some b \in R. Since R/(p) is an integral domain (as (p) is prime), the zero element in the quotient has no nonzero annihilators, implying b \in (p). But p b = 0 and b = p c for some c would imply p (p c) = 0, so p^2 c = 0. Since R is a domain, c = 0, hence b = 0. 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 \mathbb{Z}[\sqrt{-5}], the element 3 is irreducible but not prime, as it divides the product (1 + \sqrt{-5})(1 - \sqrt{-5}) = 6 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 \mathbb{[Z](/page/Z)} 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 R with identity, an ideal P is a prime ideal if P \neq R and whenever ab \in P for a, b \in R, then a \in P or b \in P; equivalently, the quotient ring R/P is an integral domain.^[18]^[19] A fundamental connection between prime elements and prime ideals arises in integral domains. Specifically, in an integral domain R, a nonzero nonunit element p \in R is prime if and only if the principal ideal (p) it generates is a nonzero prime ideal. To see this, suppose p is prime and ab \in (p), so p \mid ab; by the definition of primality, p \mid a or p \mid b, hence a \in (p) or b \in (p), showing (p) is prime. The converse follows similarly, as membership in (p) corresponds to divisibility by p.^[19] However, not all prime ideals are principal. For instance, in the polynomial ring \mathbb{Z}, the ideal (2, x) is prime because the quotient \mathbb{Z}/(2, x) \cong \mathbb{Z}/2\mathbb{Z}, which is an integral domain (in fact, a field), but (2, x) is not principal, as any single generator would need to divide both 2 and x, which is impossible in \mathbb{Z}.^[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 \mathbb{Z}, which is an integral domain, the prime elements are the associates of the positive prime numbers, such as \pm 2, \pm 3, and \pm 5.^[24] These elements satisfy the defining property: if a prime p divides a product ab in \mathbb{Z}, then p divides a or p divides b, as established by Euclid's lemma.^[25] The units in \mathbb{Z} are solely \pm 1, ensuring that these primes are non-units and nonzero.^[24] The Fundamental Theorem of Arithmetic asserts that every nonzero non-unit element in \mathbb{Z} 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 \mathbb{Z}. Since \mathbb{Z} is a principal ideal domain (PID), its prime elements coincide precisely with the irreducible elements.^[27] Extending to the Gaussian integers \mathbb{Z} = \{a + bi \mid a, b \in \mathbb{Z}\}, another integral domain, the element $1+i is a prime element.^[28] Here, the ordinary prime 2 from \mathbb{Z} ramifies, factoring as $2 = -i (1+i)^2 up to units, illustrating how prime elements behave in quadratic extensions.^[28] As \mathbb{Z} 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 k, denoted k, the ring is a principal ideal domain (PID), so every irreducible element is prime.^[29] For example, the polynomial x^2 + 1 is irreducible over \mathbb{R} and thus prime in \mathbb{R}, as it generates a maximal ideal and \mathbb{R}/(x^2 + 1) \cong \mathbb{C}, a field.^[29] A key criterion for irreducibility (and hence primality) in \mathbb{Q} is Eisenstein's criterion: if there exists a prime p \in \mathbb{Z} such that p divides all coefficients except the leading one, and p^2 does not divide the constant term, then the polynomial is irreducible over \mathbb{Q}.^[30] For instance, x^n + p x^{n-1} + \cdots + p x + p is irreducible over \mathbb{Q} for prime p \in \mathbb{Z}, as p divides the lower coefficients but p^2 does not divide p.^[30] This often establishes primality directly in the UFD \mathbb{Q}. In the polynomial ring \mathbb{Z}, which is not a PID, prime elements include certain constants and non-constant polynomials. The content of a polynomial f(x) \in \mathbb{Z} 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 \mathbb{Z} if and only if it is irreducible in \mathbb{Q}.^[31] Thus, irreducible primitive polynomials, like x^2 + 1, are prime in \mathbb{Z}. Additionally, the element x is prime, as it divides every polynomial with zero constant term, and \mathbb{Z}/(x) \cong \mathbb{Z}, an integral domain.^[32] Constant primes from \mathbb{Z}, such as 2, are also prime in \mathbb{Z}, since \mathbb{Z}/(2) \cong (\mathbb{Z}/2\mathbb{Z}), an integral domain.^[32] In multivariable polynomial rings like k[x_1, \dots, x_n] over a field k, identifying prime elements is more complex, as the ring is a UFD but not a PID for n \geq 2; prime elements are irreducible polynomials that generate height-1 prime ideals, though non-principal prime ideals of higher height exist.^[33]

References

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