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...ve rings, tertiary decompositions do, at least if the ring is [[Noetherian ring|Noetherian]]. Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutative rings. To any (two-sided) ideal, a tertiary ideal ...

...]] 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> ...

In algebra, a [[commutative ring]] ''R'' is said to be '''arithmetical''' (or '''arithmetic''') if any of th ...\mathfrak{m}</math> of ''R'' at <math>\mathfrak{m}</math> is a [[uniserial ring]] for every [[maximal ideal]] <math>\mathfrak{m}</math> of ''R''. ...

...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 ...ent in that poset. This is the case for the poset of [[prime ideal]]s of a ring, which may include the zero ideal as a [[minimal prime ideal]]. ...

...n as the [[intersection (set theory)|intersection]] of two strictly larger ideals.<ref name="m98">{{citation|title=Algebraic Geometry|volume=136|series=Trans ...es that a reducible ideal is not prime. A concrete example of this are the ideals <math>2 \mathbb Z</math> and <math>3 \mathbb Z</math> contained in <math>\m ...

In [[mathematics]], especially [[ring theory]], a '''regular ideal''' can refer to multiple concepts. ...ing theory)|ideal]] <math>\mathfrak{i}</math> in a (possibly) [[non-unital ring]] ''A'' is said to be '''regular''' (or '''modular''') if there exists an e ...

...ive Ring Theory |url=https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1 |location=Cambridge |publisher=Cambridge U For a [[regular local ring]] <math>R</math> a [[prime ideal]] <math>I</math> is perfect if and only if ...

...of fractions]], and then the conductor measures the failure of the smaller ring to be integrally closed. ...rtance in the study of non-maximal [[order (ring theory)|orders]] in the [[ring of integers]] of an [[algebraic number field]]. One interpretation of the ...

...a ring such that any [[finitely generated algebra]] over the ring is a J-1 ring. ...g]]s are J-2 rings; in fact this is part of the definition of an excellent ring. ...

...deal]] <math>\mathfrak{p}</math>, <math>L \otimes_R S</math> is a [[normal ring]]. ...on | first2=Irena |author2-link= Irena Swanson | title=Integral closure of ideals, rings, and modules | url=http://people.reed.edu/~iswanson/book/index.html ...

...s, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extract detailed information about the beha ...h>.<ref>{{harvnb|Huneke|Swanson|2006|loc=Definition 1.2.1}}</ref> For such ideals, immediately from the definition, the following hold: ...

{{Short description|Local ring in mathematics}} ...'''generalized Cohen–Macaulay ring''' is a commutative Noetherian [[local ring]] <math>(A, \mathfrak{m})</math> of [[Krull dimension]] ''d'' > 0 that sati ...

In algebra, the '''integral closure''' of an ideal ''I'' of a commutative ring ''R'', denoted by <math>\overline{I}</math>, is the set of all elements ''r ...ees (mathematician)|Rees]] that characterizes an [[analytically unramified ring]]. ...

...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 ...[ring theory]], if ''A'' is an additive subgroup of a [[ring (mathematics)|ring]] ''R'', then <math>\mathbb{I}_R(A)</math> (defined in the multiplicative s ...

In [[order theory|order-theoretic mathematics]], the '''deviation of a poset''' is an [[ordin ...o define the [[Krull dimension]] of a [[Module (mathematics)|module over a ring]] as the deviation of its poset of submodules. ...

...primes in general ring theory. For the specific usage in commutative ring theory, see also {{slink|Primary decomposition|Primary decomposition from associat ...s a type of [[prime ideal]] of ''R'' that arises as an [[annihilator (ring theory)|annihilator]] of a (prime) submodule of ''M''. The set of associated prim ...

...em''', named after [[Jacob Levitzki]], states that in a right [[Noetherian ring]], every nil one-sided ideal is necessarily [[Nilpotent ideal|nilpotent]].< ...t ''R'' satisfies the [[ascending chain condition]] on [[annihilator (ring theory)|annihilators]] of the form <math>\{r\in R\mid ar=0\}</math> where ''a'' is ...

...ith identity such that if ''I'' and ''J'' are distinct right [[ideal (ring theory)|ideal]]s then there are elements ''i'' and ''j'' such that ''i''&hairsp;'' In the [[commutative ring|commutative]] case, Gelfand rings can also be characterized as the rings su ...

...nitely generated module ''M'' over a Noetherian commutative ring ''A'' and ideals ''I'', ''J'', the following are equivalent:<ref>Takesi Kawasaki, On Falting ...hfrak{b} \supset J</math> and <math>\mathfrak b</math> [[annihilator (ring theory)|annihilates]] the [[local cohomology|local cohomologies]] <math>\operatorn ...

...a [[minimal right ideal]].</ref> Analogously the notion of a '''left Kasch ring''' is defined, and the two properties are independent of each other. ...ed [[Artinian ring]]s whose proper ideals have nonzero [[annihilator (ring theory)|annihilator]]s ''S-rings''.<ref>{{harv|Kasch|1954}}</ref><ref>{{harv|Morit ...