Maximal Ideals Research Articles

A classification of the minimal ring extensions of an integral domain

Let R be any integral domain. The minimal (commutative unital) ring extensions S of R are, up to R-algebra isomorphism, of three nonoverlapping types: (i) the domains S that contain R and are minimal ring extensions of R; (ii) the idealizations R ( + ) R / M arising from maximal ideals M of R; and (iii) the direct products R × R / M arising from maximal ideals M of R. Distinct maximal ideals of R lead to nonisomorphic idealizations (respectively direct products) in case (ii) (respectively case (iii)). If R is not a field, then distinct domains S arising in case (i) within the same quotient field of R are not isomorphic as R-algebras. If R is a field K, then the domains S arising in (i) are the minimal (necessarily algebraic) field extensions of K; in this case, distinct such fields S 1 , S 2 are K-algebra isomorphic if and only if S i = K ( α i ) where α 1 , α 2 are roots of the same irreducible polynomial in K [ X ] .

A classification of the minimal ring extensions of certain commutative rings

All rings considered are commutative with identity and all ring extensions are unital. Let R be a ring with total quotient ring T. The integral minimal ring extensions of R are catalogued via generator-and-relations. If T is von Neumann regular and no maximal ideal of R is a minimal prime ideal of R, the minimal ring extensions of R are classified, up to R-algebra isomorphism, as the minimal overrings (within T) of R and, for maximal ideals M of R, the idealizations R ( + ) R / M and the direct products R × R / M . If T is von Neumann regular, the minimal ring extensions of R in which R is integrally closed are characterized as certain overrings, up to R-algebra isomorphism, in terms of Kaplansky transforms and divided prime ideals, generalizing work of Ayache on integrally closed domains; no restriction on T is needed if R is quasilocal. One application generalizes a recently announced result of Picavet and Picavet-L'Hermitte on the minimal overrings of a local Noetherian ring. Examples are given to indicate sharpness of the results.

Projective planes over “Galois” double numbers and a geometrical principle of complementarity

The paper deals with a particular type of a projective ring plane defined over the ring of double numbers over Galois fields, R ⊗( q) ≡ GF( q) ⊗ GF( q) ≅ GF( q)[ x]/( x( x − 1)). The plane is endowed with ( q 2 + q + 1) 2 points/lines and there are ( q + 1) 2 points/lines incident with any line/point. As R ⊗( q) features two maximal ideals, the neighbour relation is not an equivalence relation, i.e. the sets of neighbour points to two distant points overlap. Given a point of the plane, there are 2 q( q + 1) neighbour points to it. These form two disjoint, equally populated families under the reduction modulo either of the ideals. The points of the first family merge with (the image of) the point in question, while the points of the other family go in a one-to-one fashion to the remaining q( q + 1) points of the associated ordinary (Galois) projective plane of order q. The families swap their roles when switching from one ideal to the other, which can be regarded as a remarkable, finite algebraic geometrical manifestation/representation of the principle of complementarity. Possible domains of application of this finding in (quantum) physics, physical chemistry and neurophysiology are briefly mentioned.

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec r (R) (resp. Max r (R), Prim r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U r (eR) = {P ∈ Spec r (R)| e ∉ P}. Let 𝒜 = ∪P∈Prim r (R) Spec r P (R), where Spec r P (R) = {Q ∈Spec r P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on 𝒜 and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if 𝒜 is a space with Zariski topology if and only if, for any Q ∈ 𝒜, Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ℬ = Max r (R) ∪ Prim r (R) (resp. 𝒞 = Prim r (R)) there is an element e in R with e 2 − e ∈ J(R) such that U = U r (eR) ∩ ℬ (resp. U = U r (eR) ∩ 𝒞), where J(R) is the Jacobson of R. (3) Max r (R) ∪ Prim r (R) is connected if and only if Max l (R) ∪ Prim l (R) is connected if and only if Prim r (R) is connected.

Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ≇ ℤ2 × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.

On the Associated Primes of Generalized Local Cohomology Modules

Let 𝔞 be an ideal of a commutative Noetherian ring R with identity and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that is contained in the union of the sets , where 0 ≤ i ≤ t. As an immediate consequence, it follows that if either is finitely generated for all i < t or is finite for all i < t, then is finite. Also, we prove that if d = pd(M) and n = dim(N) are finite, then is Artinian. In particular, is a finite set consisting of maximal ideals.

Lie algebras and algebraic groups

Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms

Ideals in constructive Banach algebra theory

We introduce the notion of a ‘q-ideal in a commutative Banach algebra, and investigate the relation between q-ideals and maximal ideals constructively.

The Jacobson semiradical over a certain semiring

The concept of the semiradical class of semirings was introduced in [3]. The purpose of this paper is to study one such semiradicals, the Jacobson semiradicals, over certain semirings. We generalize the concept of the Jacobson radical of a ring to a semiring. Some properties of the Jacobson semiradical $ JS(R) $ of the semiring $R$ parallel those of ring theory. In Section 1 we describe some preliminary definitions. In Section 2 we define regular strongly austere semimodulse. Theorem 1 characterizes a regular strongly austere semimodule in terms of a regular modular maximal subtractive left ideal. We define $JS(R)$ and derive some properties of this structure. In Section 3 we show that the $ JS(R) $ of a semiring has many representations as the intersection of left ideals. One of the more important of these is that $ JS(R) $ is the intersection of all left weakly primitive subtractive ideals. Proposition 6 characterizes a semiweekly primitive semiring in terms of a weakly primitive semiring. The interrelationships of strongly austere, weakly primitive and semiweekly primitive semirings are examined in Theorem 3. In Section 4, Proposition 7 shows that the $ JS(R) $ of a semiring with identity is the intersection of all regular maximal modular subtractive left ideals. Corollary 3 shows that $ JS(R) $ is the unique largest superfluous left ideal of $ R $. Proposition 8 shows that the class of Jacobson semiradical of semirings is closed under direct sum. We conclude with Section 5, a consideration of a certain restricted class of semirings. We show that the Jacobson semiradical for the semirings belonging to this class constitutes a semiradical class. Finally, Example 1 shows that a semiradical class need not be a radical class.

One-Sided Ideal Growth of Free Associative Algebras

Let R be a finitely generated associative algebra with unity over a finite field \({\Bbb F}_q\). Denote by a n (R) the number of left ideals J ⊂ R such that dim R/J = n for all n ≥ 1. We explicitly compute and find asymptotics of the left ideal growth for the free associative algebra A d of rank d with unity over \({\Bbb F}_q\), where d ≥ 1. This function yields a bound a n (R) ≤ a n (A d ), \(n\in{\Bbb N}\), where R is an arbitrary algebra generated by d elements. Denote by m n (R) the number of maximal left ideals J ⊂ R such that dim R/J = n, for n ≥ 1. Let d ≥ 2, we prove that m n (A d ) ≈ a n (A d ) as n → ∞.

Spectra of algebras of entire functions on Banach spaces

We obtain an explicit description of the spectrum (set of closed maximal ideals) of H b ( X ) , H_b(X), algebra of analytic functions on a Banach space X X which are bounded on bounded subsets. We show that the spectrum of H b ( X ) H_b(X) admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball B ⊂ X B\subset X are indicated.

Bipartite pseudo MV-algebras

A bipartite pseudo MV-algebra A is a pseudo MV-algebra such that A = M [ M s for some proper ideal M of A. This class of pseudo MV-algebras, denoted BP, is investigated. The class of pseudo MValgebras A such that A = M [ M s for all maximal ideals M of A, denoted BP0, is also studied and characterized.

Completions of UFDs with Semi-Local Formal Fibers

ABSTRACT Let (T, M) be a complete local ring such that |T/M| = | T |. Given a finite set of incomparable nonmaximal prime ideals C of T, we provide necessary and sufficient conditions for T to be the completion of a local UFD A, whose generic formal fiber is semilocal with maximal ideals the elements of C. We also show that, given the T above, we can find necessary and sufficient conditions for T to be the completion of a UFD, whose formal fiber over a height one prime ideal is semilocal. Communicated by I. Swanson.

On Rings whose Maximal Ideals are GP-Ideals

On Rings whose Maximal Ideals are GP-Ideals

Algebraic properties of stably numerical polynomials

We study algebraic properties of certain rings of polynomials closely related to the ring of integer-valued polynomials. We give generators and relations for the localisations at a prime. We describe the maximal and prime ideals and show that the rings considered are not Noetherian.

Maximal ideals of disjointness preserving operators

Let L and M be vector lattices with M Dedekind complete, and let L r ( L , M ) be the vector lattice of all regular operators from L into M. We introduce the notion of maximal order ideals of disjointness preserving operators in L r ( L , M ) (briefly, maximal δ-ideals of L r ( L , M ) ) as a generalization of the classical concept of orthomorphisms and we investigate some aspects of this ‘new’ structure. In this regard, various standard facts on orthomorphisms are extended to maximal δ-ideals. For instance, surprisingly enough, we prove that any maximal δ-ideal of L r ( L , M ) is a vector lattice copy of M, when L, in addition, has an order unit. Moreover, we pay a special attention to maximal δ-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators.

The Ordered Group of Invertible Ideals of a Prüfer Domain of Finite Character

Let D be a Prüfer domain, and denote by ± bℑ(D) the multiplicative group of all invertible fractional ideals of D, ordered by A ≤ B if and only if A ⊇ B. Denote by G i the value group of the valuation associated with the valuation ring D M i , where {M i } i∈I is the collection of all maximal ideals of D. In this note we prove that the natural map from ± bℑ(D) into ± b∏ i∈I G i is an isomorphism onto the cardinal sum ± b∐ i∈I G i if and only if D is h-local. As a corollary, the group of divisibility of an h-local Bézout domain is isomorphic to ± b∐ i∈I G i , the notation being as above.

The near-ring of congruence-preserving functions on an expanded group

Let V be a finite expanded group, e.g., a ring or a group. We investigate the near-ring 〈 C 0 ( V ) ;+ , ∘ 〉 of zero-preserving congruence-preserving functions on V. We obtain some information on the structure of 〈 C 0 ( V ) ;+ , ∘ 〉 from the lattice of ideals of V: for example, the number of maximal ideals of 〈 C 0 ( V ) ;+ , ∘ 〉 is completely determined by the isomorphism class of the ideal lattice of V.

Invertible modules for commutative -algebras with residue fields

The aim of this note is to understand under which conditions invertible modules over a commutative Open image in new window-algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative Open image in new window-algebra R has coherent localizations Open image in new window for every maximal ideal Open image in new window, then for every invertible R-module U, U*=π*U is an invertible graded R*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative Open image in new window-algebra has ‘residue fields’ for all maximal ideals Open image in new window if the global dimension of R* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R0. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.

Maximal Ideals in Rings

This paper gives necessary and sufficient conditions which guarantee that a ring have maximal left, right, or two-sided ideals. The relations of rings without maximal ideals to the Jacobson and Brown-McCoy radical are discussed. Examples of rings without maximal ideals are given to illustrate the theory. Connections of existence of maximal ideals and the Axiom of Choice in Zermelo-Fraenkel set theory are noted.