- principal ideal domain
*Math.*a commutative integral domain with multiplicative identity in which every ideal is principal. Also called**principal ideal ring**.[1960-65]

*Useful english dictionary.
2012.*

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**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.Principal ideal domains are thus mathematical objects which behave somewhat like the… … Wikipedia**principal ideal domain**— Math. a commutative integral domain with multiplicative identity in which every ideal is principal. Also called principal ideal ring. [1960 65] * * * … Universalium**principal ideal domain**— noun An integral domain in which every ideal is a principal ideal … Wiktionary**Structure theorem for finitely generated modules over a principal ideal domain**— In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that… … Wikipedia**Principal ideal**— In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R .More specifically: * a left principal ideal of R is a subset of R of the form R a := { r a : r in R }; * a… … Wikipedia**Principal ideal ring**— In mathematics, a principal ideal ring, or simply principal ring, is a ring R such that every ideal I of R is a principal ideal, i.e. generated by a single element a of R .A principal ideal ring which is also an integral domain is said to be a… … Wikipedia**Domain**— may refer to: General Territory (administrative division), a non sovereign geographic area which has come under the authority of another government Public domain, a body of works and knowledge without proprietary interest Eminent domain, the… … Wikipedia**Ideal class group**— In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If… … Wikipedia**Ideal (ring theory)**— In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… … Wikipedia**Ideal norm**— In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an… … Wikipedia