theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring...

different types of factor rings. Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of...

resultant ideal consists of all those polynomials whose constant coefficient is even. In any ring R, a maximal ideal is an ideal M that is maximal in the...

prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter. There is another interesting notion of maximality of ideals: Consider...

right ideal of R with {0} ⊆ K ⊆ N, then either K = {0} or K = N. N is a simple right R-module. Minimal ideals are the dual notion to maximal ideals. Many...

ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. The primitive...

abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure. Zorn's lemma...

ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean...

R} is a commutative ring and m {\displaystyle {\mathfrak {m}}} is a maximal ideal, then the residue field is the quotient ring k {\displaystyle k} = R...

after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction...

following equivalent properties: R has a unique maximal left ideal. R has a unique maximal right ideal. 1 ≠ 0 and the sum of any two non-units in R is...

topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of Grothendieck's...

appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This...

-primary. An ideal whose radical is maximal, however, is primary. Every ideal Q with radical P is contained in a smallest P-primary ideal: all elements...

Artinian ring, every maximal ideal is a minimal prime ideal. In an integral domain, the only minimal prime ideal is the zero ideal. In the ring Z of integers...

single element.) Every principal ideal domain is Noetherian. In all unital rings, maximal ideals are prime. In principal ideal domains a near converse holds:...

is a regular maximal right ideal in A. If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal. Every ring with...

principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent...

unique maximal subgroup containing e. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups...

is defined by an ideal a {\displaystyle {\mathfrak {a}}} contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced...

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168)...

A_{\mathfrak {p}}} , we can assume A {\displaystyle A} is local with the maximal ideal p {\displaystyle {\mathfrak {p}}} . Let q ⊊ p {\displaystyle {\mathfrak...

Homological conjectures Commutative ring Module (mathematics) Ring ideal, maximal ideal, prime ideal Ring homomorphism Ring monomorphism Ring epimorphism Ring...

the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nilpotent elements does not always form an ideal for...

discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R that...

Noetherian ring. Artinian Ascending chain condition for principal ideals Krull dimension Maximal condition on congruences Noetherian Proof: first, a strictly...

In mathematics, specifically ring theory, a principal ideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single...

domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain. Maximal ideals of k[V] correspond to...

that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a...

and only if the maximal ideal of R is divisorial. An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori...