Prime element

Template:Short description 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 unique factorization domains but not the same in general.

Definition

An element Template:Mvar of a commutative ring Template:Mvar is said to be prime if it is not the zero element or a unit, and for all Template:Mvar in Template:Mvar, whenever Template:Mvar divides Template:Mvar, Template:Mvar divides Template:Mvar or Template:Mvar divides Template:Mvar (that is, $p\mid ab⟹p\mid a\vee p\mid b$). With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element Template:Mvar is prime if, and only if, the principal ideal (p) generated by Template:Mvar is a nonzero prime ideal. (In an integral domain, the ideal (0) is a prime ideal, but 0 is not considered to be a prime element.) Note: References defining primality for an element $p\in R$ often restrict Template:Mvar to be an integral domain or a Euclidean domain, or may add the additional requirement that Template:Mvar is not a zero-divisor.

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

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 $ab\in I$ then either $a\in I$ or $b\in I$.

A nonzero principal ideal is prime if and only if it is generated by a prime element.

Irreducible elements

Prime elements should not be confused with irreducible elements. Recall that an element Template:Mvar of an integral domain Template:Mvar is irreducible if it is not a unit and whenever Template:Mvar, either Template:Mvar or Template:Mvar is a unit, while several non-equivalent definitions of irreducibility of varying strength exist for elements of general commutative rings (see the main article). In an integral domain, every prime is irreducible but the converse is not true in general. However, in unique factorization domains, or more generally in GCD domains, primes and irreducibles are the same.

Examples

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 $\mathbf{𝐙}[\sqrt{-5}],$ the element 3 is irreducible but not prime (as 3 divides $9=(2+\sqrt{-5})(2-\sqrt{-5})$ and 3 does not divide any factor on the right).

References

Notes

Sources

• Section III.3 of