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Concept
Irreducible element
Summary
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. It was discovered in the 19th century that the rings of integers of some number fields are not unique factorization domains, and, therefore, that some irreducible elements can appear in some factorization of an element and not in other factorizations of the same element. The ignorance of this fact is the main error in many of the wrong proofs of Fermat's Last Theorem that were given during the three centuries between Fermat's statement and Wiles's proof of Fermat's Last Theorem.
The definition can be, and usually is, extended verbatim to the elements of an arbitrary commutative ring. For a general ring , an element of is called irreducible if it is neither left-invertible nor right-invertible, and if there exists no left-invertible element together with a right-invertible element such that .
If is an integral domain, then is an irreducible element of if and only if for all , the equation implies that the ideal generated by is equal to the ideal generated by or equal to the ideal generated by . This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no zero divisors is commonly made in the definition of irreducible elements.
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element in a commutative ring is called prime if, whenever for some and in then or ) In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for unique factorization domains (or, more generally, GCD domains).
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Ontological neighbourhood
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Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
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.
Commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third.