A prime element is a distinguished type of non‑unit element in a commutative ring (or more generally in a commutative monoid or integral domain) that generalizes the familiar notion of a prime number in the ring of integers. The concept plays a central role in algebraic number theory, factorization theory, and the study of algebraic structures such as unique factorization domains and Dedekind domains.
Formal definition
Let $R$ be a commutative ring with identity, and assume that $R$ is an integral domain (i.e., it has no zero divisors). An element $p\in R$ is called a prime element if the following conditions hold:
- $p$ is neither a unit nor zero: $p eq 0$ and $p$ is not invertible in $R$.
- Whenever $p$ divides a product $ab$ with $a,b\in R$ (i.e., there exists $c\in R$ such that $ab=pc$), then $p$ divides at least one of the factors: $$ p\mid a \quad\text{or}\quad p\mid b . $$
Equivalently, the principal ideal generated by $p$, denoted $(p)=pR$, is a prime ideal of $R$. In an integral domain, the two characterizations are equivalent.
Relation to other notions
| Concept | Definition | Relationship to prime elements |
|---|---|---|
| Irreducible element | $r | |
| eq 0$ and non‑unit such that if $r=ab$ then either $a$ or $b$ is a unit. | Every prime element is irreducible, but the converse need not hold in general rings. | |
| Prime ideal | Proper ideal $\mathfrak p$ such that $ab\in\mathfrak p$ implies $a\in\mathfrak p$ or $b\in\mathfrak p$. | In a principal ideal domain (PID) or a unique factorization domain (UFD), an element $p$ is prime ⇔ $(p)$ is a non‑zero prime ideal. |
| Unique factorization domain (UFD) | Integral domain where every non‑zero non‑unit can be written uniquely (up to order and units) as a product of irreducibles. | In a UFD, the notions of prime and irreducible coincide; thus prime elements generate the “building blocks” of factorization. |
Examples
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Integers $\mathbb Z$ – The ordinary prime numbers $\pm 2, \pm 3, \pm 5,\dots$ are precisely the prime elements of $\mathbb Z$. The unit group ${\pm1}$ consists of the only units, and every non‑zero integer factors uniquely into primes up to sign.
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Polynomial ring $k[x]$ over a field $k$ – Irreducible (and hence prime) polynomials such as $x$, $x^2+1$ (over $\mathbb R$), or $x^2+1$ (over $\mathbb Q$) are prime elements. The principal ideal $(f)$ generated by an irreducible polynomial $f$ is a prime ideal.
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Gaussian integers $\mathbb Z[i]$ – Elements like $1+i$ and the rational primes congruent to $3\pmod 4$ (e.g., $3,7$) are prime elements. The ring is a Euclidean domain, hence a PID and a UFD, so prime = irreducible.
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Dedekind domains – In rings such as the ring of integers $\mathcal O_K$ of a number field $K$, prime ideals need not be principal; consequently, a “prime element” may not exist for every prime ideal. When a non‑zero prime ideal is principal, any generator of that ideal is a prime element.
Key properties
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Divisibility behavior – The defining property ensures that a prime element cannot “hide” inside a product without appearing in one of the factors. This mirrors the Euclidean property of ordinary primes.
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Generation of prime ideals – In an integral domain, $(p)$ is a prime ideal. Conversely, if an ideal generated by a single element is prime, its generator is a prime element (up to multiplication by units).
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Stability under associates – If $p$ is prime and $u$ is a unit, then $up$ is also prime; such elements are called associates.
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Localization – Under localization $S^{-1}R$ at a multiplicative set $S$, a prime element whose powers remain non‑invertible remains prime in the localized ring, provided it does not become a unit after localization.
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Non‑uniqueness in non‑UFDs – In integral domains that are not UFDs, there can exist irreducible elements that are not prime. Classical examples include certain elements in the ring $\mathbb Z[\sqrt{-5}]$.
Historical notes
The term “prime element” originates from the 19th‑century development of algebraic number theory, where mathematicians such as Ernst Kummer, Richard Dedekind, and Leopold Kronecker extended the arithmetic of integers to more general rings. The formal definition in the modern abstract algebraic language appears in early 20th‑century textbooks on ring theory.
See also
- Prime ideal
- Irreducible element
- Unique factorization domain (UFD)
- Principal ideal domain (PID)
- Euclidean domain
- Factorization in commutative rings
This entry reflects the standard definition and principal properties of prime elements as they appear in contemporary algebraic literature.