Inmathematics, specifically inabstract algebra, aprime element of acommutative ring is an object satisfying certain properties similar to theprime numbers in theintegers and toirreducible polynomials. Care should be taken to distinguish prime elements fromirreducible elements, a concept that is the same inUFDs but not the same in general.

An elementp of a commutative ringR is said to beprime if it is not thezero element or aunit and wheneverpdividesab for alla andb inR, thenp dividesa orp dividesb. With this definition,Euclid's lemma is the assertion thatprime numbers are prime elements in thering of integers. Equivalently, an elementp is prime if, and only if, theprincipal ideal(p) generated byp is a nonzeroprime ideal.^[1] (Note that in anintegral domain, the ideal(0) is aprime ideal, but0 is an exception in the definition of 'prime element'.)

Interest in prime elements comes from thefundamental 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 ofunique 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 inZ but it is not inZ[i], the ring ofGaussian integers, since2 = (1 +i)(1 −i) and 2 does not divide any factor on the right.

An idealI in the ringR (with unity) isprime if the factor ringR/I is anintegral domain. Equivalently,I is prime if whenever${\displaystyle ab\in I}$ then either${\displaystyle a\in I}$ or${\displaystyle b\in I}$.

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

Prime elements should not be confused withirreducible elements. In anintegral domain, every prime is irreducible^[2] but the converse is not true in general. However, in unique factorization domains,^[3] or more generally inGCD domains, primes and irreducibles are the same.

The following are examples of prime elements in rings:

• The integers±2,±3,±5,±7,±11, ... in thering of integersZ
• the complex numbers(1 +i),19, and(2 + 3i) in the ring ofGaussian integersZ[i]
• the polynomialsx² − 2 andx² + 1 inZ[x], thering of polynomials overZ.
• 2 in thequotient ringZ/6Z
• x² + (x² +x) is prime but not irreducible in the ringQ[x]/(x² +x)
• In the ringZ² of pairs of integers,(1, 0) is prime but not irreducible (one has(1, 0)² = (1, 0)).
• In thering of algebraic integers${\displaystyle \mathbf{Z}[\sqrt{-5}],}$ the element3 is irreducible but not prime (as 3 divides${\displaystyle 9=(2+\sqrt{-5})(2-\sqrt{-5})}$ and 3 does not divide any factor on the right).

Notes

Sources

• Section III.3 ofHungerford, 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

• Ring theory

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