Principal Ideal Domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by.

Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind rings. All Euclidean domains and all fields are principal ideal domains.

Commutative ringsintegral domainsintegrally closed domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfields

Read more about Principal Ideal Domain:  Examples, Modules, Properties

Famous quotes containing the words principal, ideal and/or domain:

    This place is the Devil, or at least his principal residence, they call it the University, but any other appellation would have suited it much better, for study is the last pursuit of the society; the Master eats, drinks, and sleeps, the Fellows drink, dispute and pun, the employments of the undergraduates you will probably conjecture without my description.
    George Gordon Noel Byron (1788–1824)

    ... probably all of the women in this book are working to make part of the same quilt to keep us from freezing to death in a world that grows harsher and bleaker—where male is the norm and the ideal human being is hard, violent and cold: a macho rock. Every woman who makes of her living something strong and good is sharing bread with us.
    Marge Piercy (b. 1936)

    In the domain of Political Economy, free scientific inquiry meets not merely the same enemies as in all other domains. The peculiar nature of the material it deals with, summons as foes into the field of battle the most violent, mean and malignant passions of the human breast, the Furies of private interest.
    Karl Marx (1818–1883)