Prime element

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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 UFDs 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 whenever Template:Mvar divides Template:Mvar for all Template:Mvar and Template:Mvar in Template:Mvar, then Template:Mvar divides Template:Mvar or Template:Mvar divides Template:Mvar. 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)Script error: No such module "Check for unknown parameters". generated by Template:Mvar is a nonzero prime ideal.[1] (Note that in an integral domain, the ideal (0)Script error: No such module "Check for unknown parameters". is a prime ideal, but 0Script error: No such module "Check for unknown parameters". 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 ZScript error: No such module "Check for unknown parameters". but it is not in Z[i]Script error: No such module "Check for unknown parameters"., the ring of Gaussian integers, since 2 = (1 + i)(1 − i)Script error: No such module "Check for unknown parameters". and 2 does not divide any factor on the right.

Connection with prime ideals

Script error: No such module "Labelled list hatnote". An ideal IScript error: No such module "Check for unknown parameters". in the ring RScript error: No such module "Check for unknown parameters". (with unity) is prime if the factor ring R/IScript error: No such module "Check for unknown parameters". is an integral domain. Equivalently, IScript error: No such module "Check for unknown parameters". is prime if whenever ab∈I then either a∈I or b∈I.

In an integral domain, a nonzero principal ideal is prime if and only if it is generated by a prime element.

Irreducible elements

Script error: No such module "Labelled list hatnote". 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

The following are examples of prime elements in rings:

  • The integers ±2Script error: No such module "Check for unknown parameters"., ±3Script error: No such module "Check for unknown parameters"., ±5Script error: No such module "Check for unknown parameters"., ±7Script error: No such module "Check for unknown parameters"., ±11Script error: No such module "Check for unknown parameters"., ... in the ring of integers ZScript error: No such module "Check for unknown parameters".
  • the complex numbers (1 + i)Script error: No such module "Check for unknown parameters"., 19Script error: No such module "Check for unknown parameters"., and (2 + 3i)Script error: No such module "Check for unknown parameters". in the ring of Gaussian integers Z[i]Script error: No such module "Check for unknown parameters".
  • the polynomials x2 − 2Script error: No such module "Check for unknown parameters". and x2 + 1Script error: No such module "Check for unknown parameters". in Z[x]Script error: No such module "Check for unknown parameters"., the ring of polynomials over ZScript error: No such module "Check for unknown parameters"..
  • 2 in the quotient ring Z/6ZScript error: No such module "Check for unknown parameters".
  • x2 + (x2 + x)Script error: No such module "Check for unknown parameters". is prime but not irreducible in the ring Q[x]/(x2 + x)Script error: No such module "Check for unknown parameters".
  • In the ring Z2Script error: No such module "Check for unknown parameters". of pairs of integers, (1, 0)Script error: No such module "Check for unknown parameters". is prime but not irreducible (one has (1, 0)2 = (1, 0)Script error: No such module "Check for unknown parameters".).
  • In the ring of algebraic integers 𝐙[−5], the element 3Script error: No such module "Check for unknown parameters". is irreducible but not prime (as 3 divides 9=(2+−5)(2−−5) and 3 does not divide any factor on the right).

References

Notes

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  1. ↑ Script error: No such module "Footnotes"., as indicated in the remark below the theorem and the proof, the result holds in full generality.
  2. ↑ Script error: No such module "Footnotes".
  3. ↑ Script error: No such module "Footnotes".

Script error: No such module "Check for unknown parameters".

Sources
  • Section III.3 of Script error: No such module "citation/CS1".
  • Script error: No such module "citation/CS1".
  • Script error: No such module "citation/CS1".
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