Extension Of Commutative Rings Research Articles

Extensions of commutative rings in which trees of prime ideals contract to trees

Une extensionA⊂B des anneaux (commutatifs) satisfait a la propriete si tout arbre dans Spec(B) couvre un arbre dans Spec(A). Il est possible qu'une extension entiere d'un anneau Noetherien ne satisfait pas a . SiA⊂B soit unei-extension satisfaisante a soit “going-up” soit “going-down”, alorsA⊂B satisfait a . Cependant, une extension d'anneaux satisfaisante a “going-up”, “going-down”, et peut etre nonunibranche dans hauteur >1. Un anneau integreA a le spectre d'un arbre si et seulement siA⊂B satisfait aP pour tout anneau integreB contenantA (resp., suranneau de BezoutB deA). De plus, si un anneau integreA n'ait pas de spectre d'un arbre mais soit localement de dimension finie, (par exemple, tout anneau integre Noetherien de dimension au moins 2), alors il existe un suranneau de BezoutB deA et un arbre sature dans Spec(B) de sorte que card=4 et l'image de a l'egard de la fleche canonique Spec(B)→Spec(A) est un ensemble sature tel que card =3 mais n'est pas d'arbre. On donne egalement des caracterisations associees des classes desi-domaines et des ai-domaines.

Launch - Installation and Propulsion Control Satellite Locator System

Installation dynamics, optimal class of thruster propulsion, specific impulse thrust for reignitable fuel motors to regulate pointing error is given for satellite parked in transfer orbit for geosynchronous (e.g. the Indian National Satellite, INSAT-1B), sunsynchronous (e.g. the Indian Earth Resource Satellite, IRS-1) orbits. Filter dynamics for slow/fast terrestrial system are obtained. Design for inter satellite link server to maximize drop outlets is given. Extension of commutative Rings for coordinated control/communication of targets with predetermined signature in global and protected environment is given. Visibility model of flexible structure space monitoring station with associated space coordinate system is given. Technique proposed is applied for information acquisition synchronization of inertial targets. Design of long range navigation in projective space and robust controller AAFM auto pilot system for hybrid reliability is given.

Cohomology of cofibred categorical groups

In this paper we study and interpret a certain non-abelian cohomology H i( B, G) , 0≤ i≤2, of a small category B with coefficients in a B -cofibred categorical group G . The work is developed in a purely abstract setting, but several examples that indicate a very close connection with algebraic and topological problems are discussed explicitly. For instance, we obtain two new interpretations for the Brauer group of a Galois extension of commutative rings: an algebraic one in terms of equivalence classes of torsors over the Galois group and a topological one in terms of homotopy classes of cross-sections for a fibration over an Eilenberg–MacLane space of type K( G,1).

Hecke actions on the K-theory of commutative rings

We prove that in the case of a Galois extension of commutative rings R ⊂- S with Galois group G, the correspondence H → K i ( S H ) ( H ≤ G) defines a cohomological G-functor. This gives a partial generalisation of results of Roggenkamp, Scott and Verschoren who consider the case of Picard groups. We use the equivalence of cohomological G-functors and Hecke actions (Yoshida, 1983) to derive some results about the structure of K-theory groups of rings of algebraic integers.

Manis valuations and Prüfer extensions. I

We call a commutative ring extension A \subset R Prüfer, if A is an R -Prüfer ring in the sense of Griffin (Can. J. Math. 26 (1974)). These extensions relate to Manis valuations in much the same way as Prüfer domains to Krull valuations. We develop a basic theory of Prüfer extensions and give some examples. In the introduction we try to explain why Prüfer extensions deserve interest from a geometric viewpoint.

Automorphisms and forms of lie algebras of cartan type

Let L be any of simple Lie algebras of Cartan type L = X(2)(m : n : Φ), X = 5, H or K, n≠1, over a field F of characteristic p > 3, and R a commutative ring extension of F. Then AutR(R⊗L)≌ Aut R(R⊗21 (m : n) : R⊗L). It follows that, all forms of L are standard and thereby are determined

On [formula omitted]p-extensions of commutative rings

We study Z p -extensions of a commutative ring R. Some general properties corresponding to the finite Galois theory by Chase, Harrison and Rosenberg are proved. After that, we consider Z p -extensions of commutative rings of characteristic p. We describe the structure of a Z p -extension and the Z p -module T( Z p , R) of the isomorphism classes of Z p -extensions of R, via Witt vectors. Results on cyclic p n -extensions of rings of characteristic p, which are already known, are also recovered by direct and elementary methods.

FINITE AUTOMORPHISM GROUPS OF REDUCED COMMUTATIVE RINGS

Abstract Definitions of normality and separability for unitary commutative ring extensions are given from the perspective of algebraic and transcendental ring extensions. Necessary and sufficient conditions are given for the set of subrings T i , R ⊆ T i ⊆ S, in a normal separable extension S: R in which the extensions S: T i are normal separable extensions, to map uniquely bijectively on the set of all subgroups of the Galois group of S: R.

Catenarity and Gelfand-Kirillov dimension in Ore extensions

If R is a commutative afline domain (for example, the coordinate ring of an irreducible algebraic variety), the sum ,of the height of any prime ideal in R and the dimension of the corresponding factor ring is the dimension of R. This implies that if Q and P are prime ideals of R with Q 2 P, any saturated chain of prime ideals from P to Q has length dim R/P dim R/Q, and so R is catenary. Schelter, Gabber, and others have proven analogous statements for some noncommutative affine rings R. In this paper we study the question of when an Ore extension of a commutative ring is catenary, giving both positive and negative results. We show that for locally finitedimensional Ore extensions of commutative affine rings, the above dimension statements are still true if we use the Gelfand-Kirillov dimension.

Unramified Kummer extensions of prime power degree

We determine, for which x∈R* the Kummer extension K(Pn√x) is unramified over K, where K is a local field of char. 0 and res.char. p, ζpn∈K, and R is the valuation ring of K. This builds on techniques of Hasse. Actually, a much more general result concerning cyclic Galois extensions of commutative rings is proved.

THE FUNDAMENTAL THEOREM OF GALOIS THEORY

For arbitrary categories and and an arbitrary functor the author introduces the notion of an -normal object and proves a general type of fundamental theorem of Galois theory for such objects. It is shown that the normal extensions of commutative rings and central extensions of multi-operator groups are special cases of -normal objects.Bibliography: 14 titles.

Unit groups and commutative ring extensions

Unit groups and commutative ring extensions

Some properties of commutative ring extensions

Some properties of commutative ring extensions

Remarks on module-finite pairs

Let R ⊊ T be an extension of commutative rings having the same identity. A. Wadsworth (10) studies the situation when R and T are integral domains, and all rings between R and T are Noetherian. In this case (R, T) is called a Noetherian pair. In a similar vein, E. Davis (4) studies normal pairs and I. Papick (8) shows when coherent pairs are Noetherian pairs.

Some Countability Conditions on Commutative Ring Extensions

Some Countability Conditions on Commutative Ring Extensions

Some countability conditions on commutative ring extensions

If S S is a finitely generated unitary extension ring of the commutative ring R R , then S S cannot be expressed as the union of a strictly ascending sequence { R n } n = 1 ∞ \{ {R_n}\} _{n = 1}^\infty of intermediate subrings. A primary concern of this paper is that of determining the class of commutative rings T T for which the converse holds—that is, each unitary extension of T T not expressible as ∪ 1 ∞ T i \cup _1^\infty {T_i} is finitely generated over T T .

Pairs of Rings with the Same Prime Ideals

There are numerous instances in which the partners in an extension of commutative rings R ⊂ T have the same prime ideals, i.e., in which Spec(R) = Spec(T). Although this equality is intended to be taken set-theoretically, the identification easily extends to the corresponding spaces endowed with their Zariski topologies (see Proposition 3.5(a)), but of course need not extend to an identification of Spec(R) and Spec(T) as affine schemes. Perhaps the most striking recent illustration of this phenomenon arises from the work of Hedstrom and Houston [14] in which R is a pseudo-valuation domain and T is a suitable valuation overring. Other examples may be found by means of the D + M construction, either in its traditional form [12, p. 560] or in the generalized situation introduced by Brewer and Rutter [5].

A characterization of a cyclic Galois extension of commutative rings

A characterization of a cyclic Galois extension of commutative rings

Kummer extensions of rings

The theory of Kummer extensions of commutative rings is constructed, generalizing the theory of Kummer extension fields. A Galois extension S/R with Abelian Galois group of degreen is called Kummerian if in the group of invertible elements of the ringR there is a cyclic subgroup of ordern such that for any , θ≠1 element 1-θ is invertible inR. Any cyclic Kummerian extension can be obtained by means of a suitable factorization of the tensor algebra over a finitely generated protectiveR -module of rank 1 (an arbitrary Kummerian extension is a tensor product of cyclic ones). The group of equivalence classes of Kummerian extensions with fixed Galois group is studied.

Brauer groups and Amitsur cohomology for general commutative ring extensions

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups H> n( S R , U) (where U means the multiplicative group) and H n( S R , Pic) and a third sequence of groups H n ( J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and H n ( J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs ( P, α) with P a rank one, projective S n-module and α an isomorphism from the coboundary of P ( in Pic S n + 1 ) to S n + 1 . (Here S n denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group B( S R ) to H 2( J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H 2( J) with Ker[ H 2( R, U)→ H 2( S, U)], where H 2( R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).