Showing papers on "Ring (mathematics) published in 1982"

Finite linear groups whose ring of invariants is a complete intersection

Abstract: The celebrated Shephard-Todd-Chevalley theorem says that for a finite linear group G operating on the «-dimensional complex vector space the ring R of invariant polynomials is a polynomial ring if and only if G is generated by pseudoreflections (g G G is a pseudoreflection if rank(g I) = 1). In this note we give a simple topological proof of the following statement : If R has m generators such that their ideal of relations is generated by m — n + s elements, then G is generated by those g £ G such that rankfe I) < j + 2. In the case s — 0 this gives a necessary condition for R to be a complete intersection. Our argument also gives a new simple proof of the \"only if\" part of the Shephard-Todd-Chevalley theorem in the case of an arbitrary ground field. Let k be a field and let G be a finite subgroup of GL(n, k). The group G acts naturally on the polynomial ring S = k[xx, . . . , xn] and we put R = Sfi to be the invariant subring of G. We say that R is a polynomial ring if R is generated by n (algebraically independent) elements, and that R is a complete intersection if R is isomorphic to k\\yl9 . . . ,yn+r]/J, where / is an ideal generated by r (= emb dim R dim R) elements. In this paper we prove the following THEOREM A. If R is a complete intersection, then G is generated by the set {g G G| rankfe 1 ) < 2} (where I is the identity matrix). The proof is based on two simple topological lemmas. We can assume that the ground field k is algebraically closed. Let ƒ: Spec(5) —• Spec(#) be the quotient morphism. Let X' and Y be the henselisations of Spec(5) at 0 and of Spec(jR) at /(O) respectively and ƒ': X* —* Y the associated morphism. Then the action of G on Spec(S) lifts to X' and/ ' is the quotient morphism. We use henselisations in order to deal with simply connected (i.e. without nontrivial étale coverings) schemes X' and Y. If char k = 0, then SpecCS) and Spec(R) are simply connected and the henseHsation is not necessary. LEMMA 1. Let Y be a simply connected scheme, Z a closed subscheme and Y = Y Z. If Y is a complete intersection and codim Z > 3, then Y is simply connected. PROOF. The proof follows from [2, X, 3.3 and 3.4]. Received by the editors August 4, 1981. 1980 Mathematics Subject Classification. Primary 14D25; Secondary 14L30. © 1982 American Mathematical Society 0273-0979/81 /0000-0330/$01.75 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 222 VICTOR KAC AND KEMCHI WATANABE REMARK 1. The conclusion of Lemma 1 holds if instead of Y to be a complete intersection and codim Z > 3, we require that Y is regular and codim Z > 2. LEMMA 2 (VINBERG). Let X be an integral scheme and G a finite subgroup of kutk{X). Let Y = X/G and f:X —> Y be the quotient morphism. For a closed point x of X let Gx denote the stabilizer of x. If Y is simply connected, then G is generated by all Gx s. PROOF. Let H be the subgroup of G generated by all Gx's; it is a normal subgroup. Then for the action of G/H on X/H any g =£ e has no closed fixed points and by [1, I, 10.11], the morphism X/H —* Y is an étale covering. By our assumption, we have G — H. PROOF OF THEOREM A. For g G G let Lg denote the subscheme of fixed points of g on X. Let L be the union of all L 's with codim L > 3, and put X — X!~L,Zf(L) and Y = Y* Z. Note that F* is a complete intersection since Spec(#) is, and Z is a closed subscheme in Y of codimension > 3. Furthermore, X is an integral scheme with the induced G-action, Y = X/G, and Y is simply connected by Lemma 1. Hence, by Lemma 2, G is generated by all Gx's, x G X. But by the definition of X, g G Gx for some x G X if and only if codim Lg<2 or, equivalently, rankfe 1 ) < 2. REMARK 2. 7? is a complete intersection for any G C GL(2, C) (F. Klein). It is not difficult to construct an example of a finite group G C £7,(3, C) generated by two matrices A1 and A2, such that rank(ylI. 7) = 2, / = 1,2, but 7? is not a complete intersection [7]. REMARK 3. Our argument together with Remark 1 gives a short topological proof of the \"only if' part of the Shephard-Todd-Chevalley theorem [3, 5] over any ground field k: If R is a polynomial ring, then G is generated by pseudoreflections. It is not difficult to show that, furthermore, Gx is generated by pseudoreflections for any x. The first author takes this opportunity to suggest the following risky conjecture: Conversely, if Gx is generated by pseudoreflections for any x, then R is a polynomial ring. If the ground field is the field C of complex numbers, the topology of Spec(7?) is better known and we can prove the following more general theorem. THEOREM B. Let G be a finite subgroup of GL(n, C) and S = C [xt,..., xn ]. IfR=S has m generators such that their ideal of relations is generated by m-n + s elements, then G is generated by those g G G such that rankfe T) < 5 + 2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINITE LINEAR GROUPS 223 PROOF. We set X* = Spec(5), Y = Spec(#)By the same argument as above, we have only to prove that Y = Y Z is simply connected if codim Z < s + 3, under our assumption. The corresponding generalisation of Lemma 1 in the complex case has been recently proved by Goresky and Macpherson [4]. REMARK 4. We do not know whether Theorem B is true for an arbitrary ground field. Note, finally, that we can strengthen Theorem A (and in a similar way, Theorem B) as follows (cf. Remark 3). THEOREM C. If R is a complete intersection, then each Gx is generated by {g£Gx\\ rankfc-7)< 2}. PROOF. Let X = Spec(£), Y = X/GX and denote by TT: X —• Y the quotient morphism. Then the morphism Y —» X/G is e'tale at n(x) by [1,1, 10.11], Hence the local ring at n(pc) E. Y is a complete intersection, and we can apply Theorem A. REMARK 5. The converse of Theorem C is false (cf. Remark 2). The authors are grateful to the organizers of the meeting in Trento, Italy (June 1981), where the authors met, for their hospitality, especially to C. Procesi who also suggested to combine the topological argument of the first author and a conjecture of the second author. Also, the authors are thankful to P. Deligne who suggested to use henselisation, and to M. Goresky and R. Macpherson who showed us the proof of the fact used to prove Theorem B. The first author thanks B. Weisfeiler for debates concerning the conjecture in Remark 3 and gratefully acknowledges the support of IHES. REFERENCES 1. A. Grothendieck, Revêtements étales et groupes fondamental (SGA, 1), Lecture Notes in Math., vol. 224, Springer-Verlag, Berlin and New York, 1971. 2. , Cohomologie locale des faisceaux cohérents et théorèmes des Lefschetz locaux et globaux (SGA, 2), North-Holland, Amsterdam, 1968. 3. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 67 (1955), 778-782. 4. M. Goresky, Letter to the first author, June 1981. 5. G. G. Shephard and J. A. Todd, Finite reflection groups, Canad. J. Math. 6 (1954), 274-304. 6. K.-L Watanabe, Invariant subrings of finite groups which are complete intersections. I. Invariant subrings of finite Abelian groups, Nagoya Math. J. 77 (1980), 89—98. 7. K.-i. Watanabe and D. Rotillon, Invariant subrings of C[X, Y, Z] which are complete intersections (in preparation). DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139 DEPARTMENT OF MATHEMATICS, NAGOYA INSTITUTE OF TECHNOLOGY, GOKISO-CHO, SHOWA-KY, NAGOYA, 466, JAPAN License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Spatial correlations in a self-repelling ring polymer

Abstract: The scattering structure function SN(q) of a large flexible ring polymer with N monomers in a good solvent is calculated as a universal expansion in the variable q2

Narrowness implies uniformity

Abstract: This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.

Finite-dimensional right duo algebras are duo

Abstract: ABsmTcr. Available examples of right (but not left) duo rings include rings without unity element which are two dimensional algebras over finite prime fields. We prove that a right duo ring with unity element which is a finite dimensional algebra over an arbitrary field K is a duo ring. This result is obtained as a corollary of a theorem on right duo, right artinian rings R with unity: left duo-ness is equivalent to each right ideal of R having equal right and left composition lengths, which is equivalent to the same property on R alone. Another result concerns algebras over a field which are semiprimary right duo rings: such an algebra is left duo provided (1) the algebra is finite dimensional modulo its radical and (2) the square of the radical is zero. These two provisions are shown to be essential by examples which are local algebras, duo on one side only.

Some examples of modules over Noetherian rings

Abstract: 1. Let n be an integer greater than zero. There exists a prime Noetherian ringR of Krull dimension + l and n a finitely generated essential extension W of a simpleR-module V such that(i) W has Krull dimension n, and(ii) W/V is n-critical and cannot be embedded in any proper of its submodules.We refer the reader to [6] for the definition and properties of Krull dimension.Theorem 1 answers questions of Jategaonkar and Goldie. Let R be a two-sidedNoetherian ring. In [7] Jategaonkar asks whether every finitely generated essentialextension of a simple R -module is artinian, and Goldie [4] asks whether a criticalR-module is necessarily compressible.The ring R is the enveloping algebra of a certain finite dimensional metabelian Liealgebra.Finitely generated, non-artinian essential extensions of simple R -modules werestudied in [8] for the case where R is a polycyclic group algebra. An example of a1-critical module which is not compressible was found independently by Goodearl [5].This example closely resembles our module W/V for the case n = 1.We note that the bounds on Krull dimension are best possible for a prime Noetherianring R of Krull dimension n + l. For, by [8, Proposition 5.5], a finitely generated essentialextension of a simple R -module can have Krull dimension at most n, while [6, Proposition6.8] states that a 1-critican n + l R-module is isomorphic to a right ideal of R and socannot have the property expressed in (ii).A simplified version of this example (the case n = 1) is to appear in [2, Chapter 7]. Iam very grateful for the hospitality of the University of Alberta where this work wascompleted.2. The example. Let k be a field of characteristic zero and X a vector space over kwith basis y,0, x

An Algebraic Description of Painted Digital Pictures

Abstract: An algebraic system for binary digital pictures has already been described, along with the definition of the four arithmetic rules. In this paper, an extension of the binary algebraic system to a 2n-valued one is first proposed. It then becomes evident that this extended algebraic system satisfies several properties including those of a ring. An example of a 2n-valued model, an eight-valued algebraic system, is introduced and applied to painted digital pictures. Pictorial operations such as multiple arrangement, enlargement, differentiation, integration, and color changes are then dealt with by the combinations of the four arithmetic rules.

Residually small varieties ofK-algebras

Abstract: We focus on those universal algebras which ring theorists call associative algebras overK, whereK is a commutative ring having an identity element. They are calledK-algebras in this paper. For fixedK there are two full varieties to consider namelyK-algebras with, or without, an identity element as a formal constant.

Constants of derivations of prime rings

Abstract: A Galois correspondence theorem is proved for any finite-dimensional Lie -algebra of outer derivations of a prime ring of positive characteristic. A theorem is proved on the existence of a locally finite ideal, in the sense of Sirsov, over the ring of constants of such a Lie -algebra. Extension and rigidity theorems are also obtained. Bibliography: 14 titles.

Quotient rings of semiprime rings with bounded index

Abstract: We say that a ring R has bounded index if there is a positive integer n such that an = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).

Metrically complete regular rings

Abstract: This paper is concerned with the structure of those (von Neumann) regular rings R which are complete with respect to the weakest metric derived from the pseudo-rank functions on R, known as the N*-metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all 80-continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group KO(R), which is proved to be an archimedean normcomplete interpolation group. Such a group has a precise representation as affine continuous functions on a Choquet simplex, from earlier work of the author and D. E. Handelman, and additional aspects of its structure are derived here. These results are then translated into ring-theoretic results about the structure of R. For instance, it is proved that the simple homomorphic images of R are right and left self-injective rings, and R is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective R-modules are determined by the isomorphism classes modulo the maximal two-sided ideals of R. As another example of the results derived, it is proved that if all simple artinian homomorphic images of R are n X n matrix rings (for some fixed positive integer n), then R is an n X n matrix ring. All rings in this paper are associative with 1, and all modules are unital right modules. For the overall theory of regular rings, we refer the reader to [2]; for the general development of Ko of regular rings as partially ordered abelian groups, and the theory of partially ordered abelian groups via their state spaces, we refer the reader to [2, 4]. In particular, these references should be consulted for more detail on definitions and concepts which are just sketched here. I. N*-completeness. Completeness of a regular ring with respect to a rank function, or with respect to a family of pseudo-rank functions, implies that the ring is right and left self-injective [2, Theorems 19.7 and 20.8], hence a considerable amount of structure theory is available for such rings [2, Chapters 9-12]. The purpose of this paper is to investigate a much broader class of regular rings, namely those which are complete with respect to the (pseudo-) metric obtained from the supremum N* of all pseudo-rank functions on the ring. In particular, all regular rings complete with respect to a family of pseudo-rank functions are N*-complete, but also, as we prove later in this section, all regular rings with bounded index of nilpotence and all 8 0-continuous regular rings are N*-complete. Received by the editors January 13, 1981 and, in revised form, April 3, 1981. 1980 Mathematics Subject Classification. Primary 16A30, 16A54, 06F20.

Cofinitely generated and cofinitely related modules

Abstract: P. VAMOS [11] has defned and studied 'finitely embedded modules' as the dual of 'finitely generated modules'. JANS [4] called them as co finitely generated modules and defined a right co-noetherian ring as dual to a right noetherian ring and investigated their properties. In this paper, we obtain a categorical justification for finitely embedded modules to be the dual of finitely generated modules, and derive some more properties of cofinitely generated modules. Then we define, as dual to 'a finitely related module', 'a cofinitely related module' and derive its properties. Throughout this paper, by a ring R we mean an associative ring with identity

Weighted expansions for canonical desingularization

Abstract: Notation.- Semigroups.- Strings.- Semigroup strings with restrictions.- Ordered semigroup strings with restrictions.- Strings on rings.- Indeterminate strings.- Indeterminate strings with restrictions.- Restricted degree and order for indeterminate strings.- Indexing strings.- Nets.- Semigroup nets with restrictions.- Ordered semigroup nets with restrictions.- Nets on rings.- Indeterminate nets.- Indeterminate nets with restrictions.- Restricted degree and order for indeterminate nets.- Prechips.- Isobars for prechips and Premonic polynomials.- Substitutions.- Substitutions with restrictions.- Coordinate nets and Monic polynomials.- Graded ring of a ring at an ideal.- Graded ring of a ring.- Graded rings at strings and nets and the notions of separatedness and regularity for strings and nets.- Inner products and further notions of separatedness and regularity for strings.- Inner products and further notions of separatedness and regularity for nets.- Weighted isobars and weighted initial forms.- Initial forms for regular strings.- Initial forms for regular strings and nets.- Protochips and parachips.- N-support of an indexing string for 2?N?6.- Prescales.- Derived prescales.- Supports of prescales.- Protoscales.- Inner products for protoscales.- Scales and isobars.- Properties of derived prescales.- Isobars for derived scales.- Isobars and initial forms for scales.- Initial forms for scales and regular nets.- Isobars for protochips.- Initial forms for protochips and monic polynomials.

Hereditariness of strong and stable radicals

Abstract: The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2, 6, 7]. All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left (right) stable if (ρ): for every ring R and every left (right) ideal I of R it follows S(I) ⊆ S(R) .

Brauer Groups in Ring Theory and Algebraic Geometry

Abstract: Generic splitting fields.- Crossed products over graded local rings.- Brauer group and diophantine geometry: A cohomological approach.- Brauer groups and class groups for a Krull domain.- Some remarks on Brauer groups of Krull domains.- Generic algebras.- Splitting rings for azumaya quaternion algebras.- Sur les decompositions des algebres a division en produit tensoriel d'algebres cycliques.- Local structure of maximal orders on surfaces.- Left ideals in maximal orders.- Brauer-Severi varieties.- On the Brauer group of surfaces and subrings of k[x,y].- The Brauer groups in complex geometry.- When is Br(X)=Br?(X)?.- Quaternionic modules over ?2 (?).- The Brauer group of a quasi affine-scheme.- A check list on Brauer groups.