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...

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...

ideals. An ideal I is a maximal ideal if it is proper and there is no proper ideal J that is a strict superset of I. Likewise, a filter F is maximal if...

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...

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...

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...

-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...

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...

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...

inverse in D, and they form an ideal M. This ideal is maximal among the (totally ordered) ideals of D. Since M is a maximal ideal, the quotient ring D/M is...

R. If R and S are commutative, M is a maximal ideal of S, and f is surjective, then f−1(M) is a maximal ideal of R. If R and S are commutative and S...

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...

catenary if all maximal chains between two prime ideals have the same length. center The center of a valuation (or place) is the ideal of elements of positive...

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...

prime ideals and the second over the maximal ideals. There is a bijection between the set of prime ideals of S−1R and the set of prime ideals of R that...

set of a ∈ K with v(a) ≥ 0, the prime ideal mv is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of Rv), the residue field kv = Rv/mv,...

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...

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...

right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of...

number field) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals. Any maximal ideal is a prime ideal or, more briefly...

the maximal ideal m {\displaystyle {\mathfrak {m}}} and M {\displaystyle M} is a finitely generated R {\displaystyle R} -module. Then all maximal regular...

a prime ideal in a polynomial ring, and the points of such an affine variety correspond to the maximal ideals that contain this prime ideal. The Zariski...

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...

generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal m, and suppose...

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

one ideal in Ck(M) (namely the kernel of φ), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of...

{Spec} (R)} is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning...