The Arithmetic of Integral Domains

Integral domains are commutative rings whose non-zero elements are closed under multiplication. If each nonzero element is a unit, the domain is called a field and is shipped off to Chap. 11 . For the domains D which remain, divisibility is a central question. A prime ideal has the property that elements outside the ideal are closed under multiplication. A non-zero element $$a\in D$$ is said to be prime if the principle ideal Da which it generates is a prime ideal. D is a unique factorization domain (or UFD) if any expression of an element as a product of prime elements is unique up to the order of the factors and the replacement of any prime factor by a unit multiple. If D is a UFD, so is the polynomial ring D[X] where X is a finite set of commuting indeterminates. In some cases, the unique factorization property can be determined by the localizations of a domain. Euclidean domains (like the integers, Gaussian and Eisenstein numbers) are UFD’s, but many domains are not. One enormous class of domains (which includes the algebraic integers) is obtained the following way: Suppose K a field which is finite-dimensional over a subfield F which, in turn, is the field of fractions of an integral domain D. One can then define the ring $$\mathcal{O}_D(K)$$ of elements of K which are integral with respect to D. Under modest conditions, the integral domain $$\mathcal{O}_D(K)$$ , will become a Noetherian domain in which every prime ideal is maximal—a so-called Dedekind domain. Although not UFD’s, Dedekind domains offer a door prize: every ideal can be uniquely expressed as a product of prime ideals (up to the order of the factors, of course).