An element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever pdividesab for some a and b in R, then p divides a or p divides b. With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element p is prime if, and only if, the principal ideal(p) generated by p is a nonzero prime ideal.[1] (Note that in an integral domain, the ideal (0) is a prime ideal, but 0 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 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.
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.
Examplesedit
The following are examples of prime elements in rings:
Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686, ISBN0-7167-1933-9, MR 1009787
Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021
December 07, 2023
prime, element, mathematics, specifically, abstract, algebra, prime, element, commutative, ring, object, satisfying, certain, properties, similar, prime, numbers, integers, irreducible, polynomials, care, should, taken, distinguish, prime, elements, from, irre. 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 which is the same in UFDs but not the same in general Contents 1 Definition 2 Connection with prime ideals 3 Irreducible elements 4 Examples 5 ReferencesDefinition editAn element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever p divides ab for some a and b in R then p divides a or p divides b With this definition Euclid s lemma is the assertion that prime numbers are prime elements in the ring of integers Equivalently an element p is prime if and only if the principal ideal p generated by p is a nonzero prime ideal 1 Note that in an integral domain the ideal 0 is a prime ideal but 0 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 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 editMain article Prime ideal An ideal I in the ring R with unity is prime if the factor ring R I is an integral domain In an integral domain a nonzero principal ideal is prime if and only if it is generated by a prime element Irreducible elements editMain article Irreducible element 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 editThe 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 Z 5 displaystyle mathbf Z sqrt 5 nbsp the element 3 is irreducible but not prime as 3 divides 9 2 5 2 5 displaystyle 9 2 sqrt 5 2 sqrt 5 nbsp and 3 does not divide any factor on the right References editNotes Hungerford 1980 Theorem III 3 4 i as indicated in the remark below the theorem and the proof the result holds in full generality Hungerford 1980 Theorem III 3 4 iii Hungerford 1980 Remark after Definition III 3 5 SourcesSection III 3 of Hungerford 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 Retrieved from https en wikipedia org w index php title Prime element amp oldid 1091158340, wikipedia, wiki, book, books, library,