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...

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...

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...

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...

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

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...

the ideal, maximally efficient, Stirling engine, for the thermal reservoirs the ratio of the heat in to the heat out is the efficiency of the ideal Carnot...

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...

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

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

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...

intersection of all prime ideals of the quotient ring. This is contained in the Jacobson radical, which is the intersection of all maximal ideals, which are the...

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...

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...

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...

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...

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)...

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

definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse...

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...

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

different ways. minimal and maximal 1.  A left ideal M of the ring R is a maximal left ideal (resp. minimal left ideal) if it is maximal (resp. minimal) among...

quadratic fields. The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant...

-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...

ring is local, and has a unique maximal right ideal, then this unique maximal right ideal is exactly J(R). Maximal ideals are in a sense easier to look...

trivial ring) contains a maximal ideal. Equivalently, in any nontrivial unital ring, every ideal can be extended to a maximal ideal. For every non-empty set...

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...