In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, more specifically ring theory In mathematics, ring theory is the study of rings; algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings , as well as an array of properties, an atomic domain is an integral domain In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 that has no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity, every non-zero non-unit In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that of which, is a finite product of irreducible elements. Atomic domains are different from unique factorization domains, because this finite decomposition of an element into irreducibles need not be unique; equivalently, not every irreducible element is a prime. Important examples of atomic domains include the class of all unique factorization domains, and all Noetherian domains. In particular, any integral domain satisfying the ascending chain condition on principal ideals (i.e. the ACCP), is a atomic domain. Despite the claim that appears in Cohn's famous paper, the converse is known to be false.[citation needed]
The term "atomic" is due to P. M. Cohn, who called a nonunit In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that of an integral domain an "atom", when it cannot be written as a product of two non-units (i.e., irreducible element).[1]
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Motivation
In this section, a ring can be viewed as merely an abstract set in which one can perform the operations of addition an multiplication; analogous to the integers.
The ring of integers (that is, the set of integers with the natural operations of addition and multiplication) satisfy many important properties. One such property is the fundamental theorem of arithmetic In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic states that any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers. For example,. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. Since a unique factorization domain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered. However, one notices that there are two aspects of the fundamental theorem of the arithmetic; that is, any integer is the finite product of prime numbers In mathematics, a prime number is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) are:, as well as that this product is unique up to rearrangement (and multiplication by units In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that). Therefore, it is also natural to ask under what conditions particular elements of a ring can be "decomposed" without requiring uniqueness. The concept of an atomic domain addresses this.
Formal definition
Let R be an integral domain In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 that has no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity. If every non-zero non-unit In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that x of R can be written as a finite product of irreducible elements, R is referred to as an atomic domain.
Special cases
In an atomic domain, it is possible that irreducible factorizations of the same nonzero nonunit have different length. An atomic domain is thus called a half-factorial domain (HFD) if any two irreducible factorization of a nonzero nonunit have the same length; i.e.,
- x = x1 x2 ... xn = y1 y2 ... ym
implies
- n = m.
Similarly, an atomic domain is called a bounded-factorial domain (BFD) if, for each nonzero nonunit x, there exists an integer N such that
- x = x1 x2 ... xn
implies
- n < N.
Every UFD is both a half-factorial domain and a bounded-factorial domain (because of the uniqueness of factorization). A bounded-factorial domain necessarily satisfies (ACCP).[citation needed]
References
- ^ P.M. Cohn, Bezout rings and their subrings
- P.M. Cohn, Bezout rings and their subrings, 1967.
Categories: Commutative algebra
