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denominators in I {\displaystyle I} , hence the name fractional ideal. The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle...

the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the...

I^{*}} . A principal fractional ideal is one of the form x R {\displaystyle xR} for some nonzero x in K. Note that each principal fractional ideal is invertible...

only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as (2, 1 + √-5). The ideal class...

an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are...

integers R. Let S be a finite set of prime ideals of R. An element x of K is an S-unit if the principal fractional ideal (x) is a product of primes in S (to...

ideals do not involve any prime ideal appearing in mf). The smaller one, Pm, is the group of principal fractional ideals (u/v) where u and v are nonzero...

fractional ideals form a group under a product. The quotient of the group of fractional ideals by the subgroup of principal ideals is then the ideal class...

denote the group of fractional ideals of K relatively prime to mf and let Pm denote the subgroup of principal fractional ideals (a) where a is near 1...

every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing...

of non-zero fractional ideals, where two such ideals are called equivalent if they are contained in the same principal fractional ideals. 2.  A Weil divisor...

{\displaystyle {\mathcal {I}}_{B}} be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed...

condition (ACC) and descending chain condition (DCC) Fractional ideal Ideal class group Radical of an ideal Hilbert's Nullstellensatz Flat module Flat map Flat...

group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain. An algebraic cycle is a higher codimension...

where IK is the group of fractional ideals of K, and PK is the subgroup of principal fractional ideals of K, that is, ideals of the form aOK where a is...

{\displaystyle s} . Such an ideal is called a principal ideal. If every ideal is a principal ideal, R {\displaystyle R} is called a principal ideal ring; two important...

are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x. The principal ideal RxR is written...

cover general valuation rings and Krull rings. To this day, Krull's principal ideal theorem is widely considered the single most important foundational...

ideal b {\displaystyle \mathbf {b} } such that the product ideal a b {\displaystyle \mathbf {ab} } is a principal ideal." Two (integral or fractional)...

received a special name, such as normal subgroup for groups and two-sided ideals for rings. Kernels allow defining quotient objects (also called quotient...