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...
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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...
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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...
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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...
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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...
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appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This...
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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...
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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...
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-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...
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Krull's theorem (redirect from Existence of maximal ideals)
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...
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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...
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Valuation ring (section Ideals in valuation rings)
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...
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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...
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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...
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Glossary of commutative algebra (redirect from Saturated ideal)
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...
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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...
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Localization (commutative algebra) (redirect from Localization at a prime ideal)
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...
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Valuation (algebra) (redirect from Maximal ideal of a valuation)
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,...
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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...
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Krull dimension (redirect from Height of an ideal)
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...
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ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. The primitive...
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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...
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Commutative ring (section Principal ideal domains)
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...
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the maximal ideal m {\displaystyle {\mathfrak {m}}} and M {\displaystyle M} is a finitely generated R {\displaystyle R} -module. Then all maximal regular...
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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...
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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...
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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...
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single element.) Every principal ideal domain is Noetherian. In all unital rings, maximal ideals are prime. In principal ideal domains a near converse holds:...
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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...
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Spectrum of a ring (redirect from Prime ideal topology)
{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...
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