Ring ideal generated by a single element of the ring
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In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in
The remainder of this article addresses the ring-theoretic concept.
In mathematics, specifically ring theory, a principalideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single...
In mathematics, a principalideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element...
In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The...
denominators in I {\displaystyle I} , hence the name fractional ideal. The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle...
In mathematics, the principalideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the...
algebra, the structure theorem for finitely generated modules over a principalideal domain is a generalization of the fundamental theorem of finitely generated...
{\displaystyle s} . Such an ideal is called a principalideal. If every ideal is a principalideal, R {\displaystyle R} is called a principalideal ring; two important...
fractional ideals of the ring of integers of K, and PK is its subgroup of principalideals. The class group is a measure of the extent to which unique factorization...
and the whole ring R. Every maximal ideal is in fact prime. In a principalideal domain every nonzero prime ideal is maximal, but this is not true in...
nonprincipal otherwise. By the principalideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This...
modules over principalideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principalideal domain: the primary...
principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals...
is a Euclidean domain, G is a principalideal domain, which means that every ideal of G is principal. Explicitly, an ideal I is a subset of a ring R such...
prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principalideal theorem...
algebra, a discrete valuation ring (DVR) is a principalideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R...
maximal ideals are the principalideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principalideal domain....
congruences) is true over every principalideal domain. It has been generalized to any ring, with a formulation involving two-sided ideals. The earliest known statement...
system for details. More generally, linear algebra is effective on a principalideal domain if there are algorithms for addition, subtraction and multiplication...
dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principalideal domain that is not a field has Krull dimension 1. A local ring has...
linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental...
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