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...]] over ''A''.<ref name="Ref_">Goldman domains/ideals are called G-domains/ideals in (Kaplansky 1974).</ref> They are named after [[Oscar Goldman (mathematic ...]]s are [[maximal ideal|maximal]] although there are infinitely many prime ideals.<ref name="Ref_a">Kaplansky, p. 13</ref> ...

...n as the [[intersection (set theory)|intersection]] of two strictly larger ideals.<ref name="m98">{{citation|title=Algebraic Geometry|volume=136|series=Trans ...section is <math>6 \mathbb Z</math>, and <math>6 \mathbb Z</math> is not a prime ideal. ...

...e conductor is that it measures the failure of unique factorization into [[prime ideal]]s. ...en that there are ideals of ''A'' contained in the conductor which are not ideals of ''B''.) ...

{{short description|Prime ideal that is an annihilator a prime submodule}} ...{mvar|M}} (word play between the notation and the fact that an associated prime is an ''annihilator'').<ref>{{cite journal|last=Picavet |first= Gabriel|tit ...

...eal''' is a non-zero left ideal of ''R'' containing no other non-zero left ideals of ''R'', and a '''minimal ideal''' of ''R'' is a non-zero ideal containing ...[prime ideal]]s of a ring, which may include the zero ideal as a [[minimal prime ideal]]. ...

...nal |first=Otto |last=Forster |title=Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring|journal=Mathematische Zeitschrift |volume=84|pag *<math>\mathfrak{p}</math> a [[prime ideal]] of <math>R</math>. ...

...braic number theory]], the '''Dedekind–Kummer theorem''' describes how a [[prime ideal]] in a [[Dedekind domain]] factors over the domain's [[Integral closu ...inimal polynomial for <math>\alpha</math> over <math>\Z[x]</math>. For any prime <math>p</math> not dividing <math>[\mathcal O_K : \Z[\alpha]]</math>, write ...

...h>.<ref>{{harvnb|Huneke|Swanson|2006|loc=Definition 1.2.1}}</ref> For such ideals, immediately from the definition, the following hold: *''J'' and ''I'' have the same radical and the same set of minimal prime ideals over them<ref>{{harvnb|Huneke|Swanson|2006|loc=Lemma 8.1.10}}</ref> (the co ...

The intersection of an arbitrary collection of ideals in <math>X</math> is again an ideal and furthermore, <math>X</math> is clea ...ood of the origin is a solid subset of the continuous dual space <math>X^{\prime}</math>; ...

{{Short description|Result concerning ideals of commutative rings}} ...]] ''R'' is contained in a [[Union (set theory)|union]] of finitely many [[prime ideal]]s ''P''_''i'''s, then it is contained in ''P''<sub>''i''</ ...

*Abian, A., Amin, W. A. (1990) "Existence of prime ideals and ultrafilters in partially ordered sets", Czechoslovak Math. J., 40: 159 *Niederle, J. (2006) "Ideals in ordered sets", [[Rendiconti del Circolo Matematico di Palermo]] 55: 287– ...

...on (set theory)|inclusion]]. The '''ascending chain condition on principal ideals''' (abbreviated to '''ACCP''') is satisfied if there is no infinite strictl ...ions uses [[Euclid's lemma]], which requires factors to be [[prime element|prime]] rather than just irreducible. Indeed, one has the following characterizat ...

*For each prime ideal <math>\mathfrak{p}</math> of <math>\widehat{A}</math> that is not <ma ...at the localization <math>A_\mathfrak{p}</math> is Cohen–Macaulay for each prime ideal <math>\mathfrak{p} \ne \mathfrak{m}</math>. ...

...s. {{harvtxt|Hasse|1926|loc=p.6}} uses "Strahl" to mean a certain group of ideals defined using positivity conditions, and uses "Strahlklasse" to mean a cose ==Ray class fields using ideals== ...

...''L'' of the [[residue field]] <math>\kappa(\mathfrak{p})</math> of any [[prime ideal]] <math>\mathfrak{p}</math>, <math>L \otimes_R S</math> is a [[normal ...on | first2=Irena |author2-link= Irena Swanson | title=Integral closure of ideals, rings, and modules | url=http://people.reed.edu/~iswanson/book/index.html ...

...prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed. ...ional]] (i.e., the completion is equidimensional.). Then two ''m''-primary ideals <math>I \subset J</math> have the same integral closure if and only if they ...

* the [[set-theoretic complement]] of a [[Prime ideal|prime]] [[Ideal (ring theory)|ideal]] in a commutative ring; * An ideal ''P'' of a commutative ring ''R'' is prime if and only if its complement {{nowrap|''R'' \ ''P''}} is multiplicatively ...

...t subsemigroup of ''S'' in which ''T'' is an [[Semigroup#Subsemigroups and ideals|ideal]].{{sfn|Mikhalev|Pilz|2002|loc=p.30}} Such an idealizer is given by Often, when right or left ideals are the additive subgroups of ''R'' of interest, the idealizer is defined m ...

...=0092-7872}}</ref><ref>{{Cite web|title=algebraic geometry - When does the prime spectrum deformation retract into the maximal spectrum?|url=https://math.st ...

For a [[regular local ring]] <math>R</math> a [[prime ideal]] <math>I</math> is perfect if and only if <math>R/I</math> is [[Cohe ...wise. Macaulay used [[Hilbert function]]s to define his version of perfect ideals. ...