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Concept
Prime element
Résumé
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.
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 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. (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 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 element
Prime elements should not be confused with irreducible elements. 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.
The following are examples of prime elements in rings:
The integers ±2, ±3, ±5, ±7, ±11, .
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Publications associées (1)
Algebraic Divisibility Sequences Over Function Fields
In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.
Australian Mathematical Society2012Concepts associés (26)
Anneau commutatif
Un anneau commutatif est un anneau dans lequel la loi de multiplication est commutative. L’étude des anneaux commutatifs s’appelle l’algèbre commutative. Un anneau commutatif est un anneau (unitaire) dans lequel la loi de multiplication est commutative. Dans la mesure où les anneaux commutatifs sont des anneaux particuliers, nombre de concepts de théorie générale des anneaux conservent toute leur pertinence et leur utilité en théorie des anneaux commutatifs : ainsi ceux de morphismes d'anneaux, d'idéaux et d'anneaux quotients, de sous-anneaux, d'éléments nilpotents.
Irreducible element
In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements. The irreducible elements are the terminal elements of a factorization process; that is, they are the factors that cannot be further factorized. The irreducible factors of an element are uniquely defined, up to the multiplication by a unit, if the integral domain is a unique factorization domain.
Prime element
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. 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 p divides ab for some a and b in R, then p divides a or p divides b.
Séances de cours associées (5)
Domaines Intégraux et Groupes AbeliensMATH-310: Algebra
Discute des domaines intégraux, des groupes abéliens, de l'inversibilité, des diviseurs zéro, des éléments premiers et de la classification des groupes.
Propriétés du champ : Irréductibilité et unitésMATH-310: Algebra
Couvre les propriétés des champs, y compris l'irréductibilité et les unités dans les polynômes.
Anneaux et champsMATH-310: Algebra
Explore les anneaux, les champs, les idéaux et leurs propriétés dans les structures algébriques.