pinecone/core-2020-05-10-deduplication huggingface dataset:
Generic Module Theory

AbstractThe process of restricting modules to cyclic shifted subgroups is a fundamental technique in the modular representation theory of elementary abelianp-groups. IfEis elementary abelian ofp-rankrandkis an algebraically closed field of characteristicp, then each point inkr−{0} determines a cyclic shifted subgroup. Because the restriction of akE-module to this shifted subgroup depends only upon the corresponding point in projective space, it is often convenient to work with2kr−1instead ofkr−{0}. Roughly speaking, this paper shows that ifVis an irreducible subvariety of3kr−1andMis akE-module, then for almost all points inVthe direct sum decomposition ofMis the same upon restriction; moreover, this decomposition is completely determined by the behavior ofMupon restriction to the cyclic shifted subgroup corresponding to the generic point ofV. A similar idea provides a stratification of the rank variety ofMinto a disjoint union of locally closed subspaces. The closures of these subspaces are then described in terms of deformations of modules over a group of orderp
Algebraic and Turing Separability of Rings

AbstractLetFbe a countable field/ring. Then aweak presentationofFis an injective homomorphism fromFinto a field/ring whose universe is2such that all the field/ring operations are translated by total recursive functions. Given two recursive integral domainsR1andR2with quotient fieldsF1andF2respectively, we investigate under what circumstances there exists a weak presentation of the fieldF1F2such that the images ofR1andR2belong to two different recursively enumerable (r.e.) Turing degrees. In many cases we succeed in giving a completely algebraic necessary and sufficient condition for the “Turing separation” described above. More specifically, under some conditions, we can make the images ofR1andR2be of arbitrary r.e. degrees. The algebraic condition is a generalization of the notion of algebraic field separability. As a result of our investigation, we also show that3has an r.e. weak presentation as a ring which is not a weak presentation of4as a field
The Bernstein Problem in Dimension 6

AbstractThe solution of the Bernstein problem in the regular and exceptional cases, in all dimensionsn, was made by Yu. Lyubich. A. Grishkov proved that there are no nonregular nonexceptional nuclear Bernstein algebras of type (4,2) with stochastic realization and therefore the Bernstein problem of type (4,2) was completely solved by the present author (J. Algebra, to appear). The aim of this paper is to describe explicitly all simplicial stochastic nonexceptional nonregular Bernstein algebras of type (3,3). Since every nonregular nonexceptional Bernstein algebra of dimension 6 is either of type (4,2) or of type (3,3), the Bernstein problem in dimension 6 is completely solved in this paper
Pro-affine Algebras, Ind-affine Groups and the Jacobian Problem

AbstractThe purpose of the paper at hand is first to develop a theory of pro-affine algebras and their dual objects, ind-affine varieties and groups, and second to apply the theory to the well-known Jacobian problem, which asks whether or not a polynomial endomorphism of the complex affinen-space with nowhere vanishing Jacobian determinant is in fact an automorphism (cf. [3], for instance). Our main result, Theorem 5.3, says the affirmative answer obtains if the natural map from the automorphisms to the endomorphisms as above is a closed immersion of ind-affine varieties.Much earlier, I. R. Shafarevich in [7, 8] introduced the notion of “infinite-dimensional varieties and groups” and investigated the structure of the automorphism group of the affinen-space. Those are the ind-affine varieties and groups in our terminology. While we owe much to Shafarevich's two papers just cited, we have found some minor flaws in his theory as discussed in the main text below (also cf. [4]). More importantly, since Shafarevich's starting point is ind-affine varieties, his rings of functions are without nonzero nilpotent elements to begin with. We have found this treatment inadequate for our purpose of application to the Jacobian problem, because certain ideals naturally associated with endomorphisms with Jacobian determinant 1 and with automorphisms arenotknown to agree with their respective radicals. Indeed, the premises of our main result, Theorem 5.3, are satisfied and the affirmative answer to the Jacobian problem follows,oncethe ideals associated with automorphisms are known to be radical ideals. (See (5.4) below, as well as [2].)So, we have undertaken to build our theory starting from the pro-affine algebras. This was, in a sense, already taken up by Abe and Takeuchi [1] recently, but we had to redo it our way because their purpose and viewpoints much differ from ours. The theory we present herein is still rudimentary, and it is hoped that a further development of the theory will ultimately settle the Jacobian problem.The notion of linearly compact rings and modules, as in the 1953 paper [9], is used in essential ways in our paper. For apparent lack of recent references, we have added a short Appendix dealing with the subject
Normal Bands and Their Inverse Semigroups of Bicongruences

AbstractIsomorphisms between quotient algebras of a given algebra form, under an appropriate multiplication, an inverse semigroup here called the inverse semigroup of bicongruences. This paper determines those normal bands whose inverse semigroups of bicongruences belong to special classes of inverse semigroups and finds conditions on the structure semilattices, rectangular band components, and linking homomorphisms of two normal bands with isomorphic inverse semigroups of bicongruences, under which the bands themselves are isomorphic or anti-isomorphic, or related in some looser manner. Some results are also given for the wider class of general bands
On Quasi-Harada Rings

AbstractM. Harada (“Ring Theory, Proceedings of 1978 Antwerp Conference,” pp. 669–690, Dekker, New York, 1979) studied the following two conditions:(*) Every non-small leftR-module contains a non-zero injective submodule.(*)* Every non-cosmall rightR-module contains a non-zero projective direct summand.K. Oshiro (Hokkaido Math. J.13, 1984, 310–338) further studied the above conditions, and called a left artinian ring with (*) a left Harada ring (abbreviated left H-ring) and a ring satisfying the ascending chain condition for right annihilator ideals with (*)* a right co-Harada ring (abbreviated right co-H-ring). K. Oshiro (Math. J. Okayama Univ.31, 1989, 161–178) showed that left H-rings and right co-H-rings are the same rings. Here we are particularly interested in the following characterization of a left H-ring given in Harada's paper above: A ringRis a left H-ring if and only ifRis a perfect ring and for any left non-small primitive idempotenteofRthere exists a non-negative integertesuch that(a)RRe/Sk(RRe) is injective for anyk∈{0,…,te} and(b)RRe/Ste+1(RRe) is a small module, whereSk(RRe) denotes thek-th socle of the leftR-moduleRe.This characterization implies(+)Ste+1(RRe) is a uniserial leftR-module for any left non-small primitive idempotentein a left H-ringR.In this paper, we generalize left H-rings by removing (+). Concretely, since a left H-ringRis also characterized by the statement thatRis left artinian and for any primitive idempotentgofRthere exist a primitive idempotentegofRand a non-negative integerkgsuch that the injective hull of the leftR-moduleRg/Jgis isomorphic toRe/Skg(RReg), whereJis the Jacobson radical ofRR, we define a more general class of rings by the condition that for any primitive idempotentgofRthere exists a primitive idempotenteofRsuch that the injective hull of the leftR-moduleRg/Jgis isomorphic toRe/{x∈Re|gRx=0}, and call it a left quasi-Harada ring (abbreviated left QH ring). In Section 1 we characterize a left QH ring by generalizing the characterizations of a left H-ring given by M. Harada and K. Oshiro. We also consider the weaker rings, left QF-2 rings and right QF-2* rings. K. Oshiro (Math. J. Okayama Univ.32, 1990, 111–118) described the connection between left H-rings and QF rings. In Section 2 we describe the connection between two-sided QH rings and QF rings
CoproductMV-Algebras, Nonstandard Reals, and Riesz Spaces

AbstractUp to categorical equivalence,MV-algebras are unit intervals of abelian lattice-ordered groups (for short,l-groups) with strong unit. While the property of being a strong unit is not even definable in first-order logic,MV-algebras are definable by a few simple equations. Accordingly, such notions as ideals and coproducts are definable for anyMV-algebraAas particular cases of the general algebraic notions. The radical RadAis the intersection of all maximal ideals ofA. AnMV-algebraAis said to be local iff it has a unique maximal ideal. Then, by Hoelder's theorem, the quotientA/RadAis isomorphic to a subalgebra of the real unit interval [0,1]. Using nonstandard real numbers we give a concrete representation of those totally orderedMV-algebrasAwhich are isomorphic to the coproduct ofA/RadAand 〈RadA〉, the latter denoting the subalgebra ofAgenerated by its radical. As an application, using several categorical equivalences we describe theMV-algebraic counterparts of Riesz spaces, also known as vector lattices
On Finite Circular Planar Nearrings

AbstractThe main subject of this paper is to characterize finite circular Ferrero pairs, the tool for constructing finite circular planar nearrings. A general characterization of finite circular Ferrero pairs is given. In particular, a generalized version of Modisett's characterization (see J. R. Clay “Nearrings: Geneses and Applications,” pp. 68–75, Oxford Univ. Press, Oxford, 1992) is presented for finite circular Ferrero pairs (N,Φ) with cyclic Φ. We show that the fixed point free group of automorphisms Φ of a finite circular Ferrero pair (N,Φ) is metacyclic. Finally, two questions on the existence of nonabelian circular planar nearrings are answered by a general construction method and some basic examples
Skew Polynomial Rings with Binomial Relations

AbstractIn this paper we continue the study of a class of standard finitely presented quadratic algebrasAover a fixed fieldK, called binomial skew polynomial rings. We consider some combinatorial properties of the set of defining relationsFand their implications for the algebraic properties ofA. We impose a condition, called (∗), onFand prove that in this caseAis a free module of finite rank over a strictly ordered Noetherian domain. We show that an analogue of the Diamond Lemma is true for one-sided ideals of a skew polynomial ringAwith condition (∗). We prove, also, that if the set of defining relationsFis square free, then condition (∗) is necessary and sufficient for the existence of a finite Groebner basis of every one-sided ideal inA, and for left and right Noetherianness ofA. As a corollary we find a class of finitely generated non-commutative semigroups which are left and right Noetherian
Convex Orderings in Affine Root Systems, II

AbstractThis paper contains several results concerning convex orderings in root systems. These results complete the analysis developed in earlier papers by the author. We obtain, in the finite case, a combinatorial characterization of reflections, longest elements in standard parabolic subgroups of the Weyl group of the root system, and of Coxeter elements. In the affine case, a refinement of the main construction done earlier by the author (J. Algebra172, 1995, 613–623) is described; moreover, infinite periodic compatible sets are classified
On π-Separable Groups

AbstractWe generalize Lemmas 7.3–7.7 from the Odd Order paper. Those lemmas we extend fromp-solvable groups to π-separable groups, where π is an arbitrary set of primes. Note that even in the case π={p} some of our results present proper generalizations. In some proofs we use the solvability of groups of odd order (in the case |π|<3 our results do not depend on the solvability of groups of odd order). We give also very short (and, sometimes, new) proofs of some related known results, for example, Thompson'sA×B-Lemma. Corollary 3 extends Lemma 8.2 of W. Feit and J. G. Thompson (Pacific J. Math.13, 1963, 775–1029) fromp-groups to solvable groups
Rings of Operators on Modules over Commutative Rings and Their Right Ideals

AbstractSuppose that there is an inclusion ofk-algebrasR⊆E⊆EndkMwithRcommutative andEnon-commutative. We introduce and impose conditions under which the finitely generated essential right ideals ofEmay be classified in terms ofk-submodules ofM. This yields a classification of the domains Morita equivalent toEwhenEis a Noetherian domain. For example, a special case of our results is:THEOREM.Let R be a commutative Noetherian k-algebra which is domain. Let E be a simple Ore extension of R of the form R[x,x−1;σ]or R[x;δ] (in the latter case we must also assume R⊃3).Then,for a certain sublatticeLof the lattice of k-submodules of R:(a)Every non-zero right ideal of E is isomorphic to one of the form[formula]for some V∈L.(b)Every domain Morita equivalent to E is isomorphic to[formula]for some V∈L.Conversely,if R is Dedekind,then E(V)is Morita equivalent to E,for V∈L
Centralp-Extensions of (p,p,…,p)-Type Galois Groups

AbstractLetpbe a prime number,Ka field with characteristic notpand containing thepth roots of unity, andE/Kan abelian exponentpGalois extension. We prove explicit formulas for the construction of fieldsN/Kwith Galois group a centralp-extension of Gal(E/K). These formulas do not require the solution of a linear system of equations in the field extension, as do the formulas of Massy [J. Algebra109(1987), 508–535]. In our study we develop generalizations of theorems of Serre and Fröhlich on representing obstructions to embedding problems and theorems of Crespo on explicitly constructing solution fields
Two Sided Tilting Complexes for Green Orders and Brauer Tree Algebras

AbstractWe give an explicit two sided tilting complex between two Green orders having the same structural data as they were defined by K. W. Roggenkamp. (1992,Comm. Algebra20, 1715–1734; and 1994,in“Finite Dimensional Algebras and Related Topics,” pp. 265–276, Kluwer Academic, Dordrecht/Norwell, MA). This yields an explicit two sided tilting complex between two Brauer tree algebras over the same field associated to trees with the same number of edges and the same exceptional multiplicity. The present work also gives a certain generalization of a result of Gabriel and Riedtmann (1979,Comment. Math. Helv.54, 240–287)
Remarks on the Jacobian Conjecture

AbstractLetF≔(F1,…,Fn)∈(C[X1,…,Xn])nwith det(J(F))∈C* and letMi(Xi,Y)=mi0(Y)+mi1(Y)Xi+···+midi(Y)Xidi∈C[Xi,Y]≔C[Xi,Y1,…,Yn] be the minimal polynomial ofFoverC(Xi). We prove thatmi0(Y),…,midi(Y) have no common zeros inCn. As a direct consequence, we obtain flatness ofC[F,Xi] overC[F] for everyi. As applications, we obtain simple algebraic proofs of the following two known results: (i) A birational polynomial map fromCnintoCnwith det(J(F))∈C* is actually an automorphism; (ii) an injective polynomial map fromCnintoCnis also an automorphism
Asymptotic Results for Primitive Permutation Groups

AbstractWe prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostncμ(n), where μ(n) denotes the maximal exponent occurring in the prime factorization ofn. This result is applied to investigating maximal subgroup growth of infinite groups. We then proceed by showing that if the point-stabilizerGαof a primitive groupGof degreendoes not have the alternating group Alt(d) as a section, then the order ofGis bounded by a polynomial inn. This result extends a well-known theorem of Babai, Cameron and Pálfy. It is used to prove, for example, that ifHis a subgroup of indexnin a groupG, andHis a product ofbcyclic groups, then |G:HG|≤ncwherecdepends onb
Narrow Lie Algebras: A Coclass Theory and a Characterization of the Witt Algebra

AbstractIn this paper we examine some narrowness conditions for Lie algebras over a fieldFof characteristic zero. In particular we show that the natural analogs of the main coclass conjectures forp-groups hold in the context of N-graded Lie algebrasLwhich are generated by their first homogeneous componentL1. While Lie algebras of finite coclass need not be soluble, we show that the positive part of the Witt algebra DerF[x] is the only non-soluble N-graded Lie algebraLof coclass 1 which is generated byL1andL2
The Core of a Module over a Two-Dimensional Regular Local Ring

AbstractThis paper explicitly determines the core of a torsion-free, integrally closed module over a two-dimensional regular local ring. It is analogous to a result of Huneke and Swanson which determines the core of an integrally closed ideal. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two-dimensional regular local ring is the product of the module and a certain Fitting ideal of the module. The technical tools used are quadratic transforms and Buchsbaum–Rim multiplicity
Lifting Group Representations to Maximal Cohen–Macaulay Representations

AbstractAuslander announced the following result: ifRis a complete local Gorenstein ring then every finitely generatedR-module has a minimal (in the sense of4maximal Cohen–Macaulay approximation. In this paper we give a non-commutative version of Auslander's result and, in particular, show that ifRis as above and ifGis a finite group then any finitely generated representation ofGoverRhas a lifting to a representation in a maximal Cohen–Macaulay module with properties analogous to those of Auslander's approximations. WhenGis trivial, we recover Auslander's approximations. We use such a lifting to construct what we call generalized Teichmüller invariants. These will be given by a canonical embedding ofGLn(Z/(p)) intoGLm( caron p) (for somem≥n) wherepis a prime whenn=1,mwill be 1, and we get the usual Teichmüller sectionZ/(p)*→Z*p. Our proof has three ingredients. These are a version of3(see Corollaries 5.4 and 6.4), our result guaranteeing the existence of Gorenstein injective envelopes7Theorem 6.1] and a duality for non-commutative rings which generalizes Matlis duality
Representations of Finite-Dimensional Hopf Algebras

AbstractLetHdenote a finite-dimensional Hopf algebra with antipodeSover a field k. We give a new proof of the fact, due toOS, thatHis a symmetric algebra if and only ifHis unimodular andS2is inner. IfHis involutory and not semisimple, then the dimensions of all projectiveH-modules are shown to be divisible by chark. In the case where k is a splitting field forH, we give a formula for the rank of the Cartan matrix ofH, reduced modchark, in terms of an integral forH. Explicit computations of the Cartan matrix, the ring structure ofG0(H), and the structure of the principal indecomposable modules are carried out for certain specific Hopf algebras, in particular for the restricted enveloping algebras of completely solvablep-Lie algebras and ofsl(2,k)
Verification of Dade's Conjecture for Janko GroupJ3

AbstractIn [3] Dade made a conjecture expressing the numberk(B,d) of characters of a given defectdin a givenp-blockBof a finite groupGin terms of the corresponding numbersk(b,d) for blocksbof certainp-local subgroups ofG. Several different forms of this conjecture are given in [5].Dade claims that the most complicated form of this conjecture, called the “Inductive Conjecture 5.8” in [5], will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive conjecture for all covering groups of the third Janko groupJ3(in the notation of the Atlas [1]). This is one step in the inductive proof of the conjecture for all finite groups.Certain properties ofJ3simplify our task. The Schur Multiplier ofJ3is cyclic of order 3 (see [1, p. 82]). Hence, there are just two covering groups ofJ3, namelyJ3itself and a central extension 3·J3ofJ3by a cyclic groupZof order 3. We treat these two covering groups separately.The outer automorphism group Out(J3) ofJ3is cyclic of order 2 (see [1, p. 82]). In this case Dade affirms in [5, Section 6] that the inductive conjecture forJ3is equivalent to the much weaker “Invariant Conjecture 2.5” in [5]. Furthermore, Dade has proved in [6] that this Invariant Conjecture holds for all blocks with cyclic defect groups. The Sylowp-subgroups ofJ3are cyclic of orderpfor all primes dividing |J3| except 2 and 3. So we only need to verify the Invariant Conjecture for the two primesp=2 andp=3. We do that in Theorem 2.10.1 and Theorem 3.6.1 below.The group Out(3·J3|Z) of outer automorphisms of 3·J3centralizingZis trivial (see [1, p. 82]). In this case Dade affirms that the Inductive Conjecture is equivalent to the “Projective Conjecture 4.4” in [5]. Again, Dade has shown in [6] that this projective conjecture holds for all blocks with cyclic defect groups. So we only need to verify the projective conjecture forp=2 andp=3. We do that in Theorem 4.4.2 and Theorem 4.3.1.For the above reasons the four Theorems 2.10.1, 3.6.1, 4.4.2, and 4.3.1 below are sufficient to prove the inductive form of Dade's conjecture for all covering groups ofJ3.This paper is my Ph.D. thesis. It would not have been possible without the help and infinite patience of my advisor Everett Dade
Integral Semihereditary Orders, Extremality, and Henselization

AbstractIn this paper, we study integral semihereditary orders over a valuation ring in a finite-dimensional simple Artinian ring. In the first section we prove that such orders are extremal. Consequently, in a central division algebra admitting a total valuation ring, the intersection of all the conjugates of the total valuation ring is the unique integral semihereditary order over the center of the total valuation ring. In the second section we characterize, up to conjugacy, integral semihereditary orders over a Henselian valuation ring. In the last section we show that an integral orderRover an arbitrary valuation ringVis semihereditary iff its Henselization,R⊗VVh, whereVhis the Henselization ofV, is a semihereditaryVh-order. In this case, there is an inclusion preserving bijective correspondence between semihereditaryV-orders insideRand semihereditaryVh-orders insideR⊗VVh
Graphs and Separability Properties of Groups

AbstractA groupGis LERF (locally extended residually finite) if for any finitely generated subgroupSofGand for anyg∉Sthere exists a finite index subgroupS0ofGwhich containsSbut notg. Using graph-theoretical methods we give algorithms for constructing finite index subgroups in amalgamated free products of groups with good separability properties. We prove that a free product of a free group and a LERF group amalgamated over a cyclic subgroup maximal in the free factor is LERF. The maximality condition cannot be removed, because adjunction of roots does not preserve property LERF. We also give short proofs of some old theorems about separability properties of groups, including a theorem of Brunner, Burns, and Solitar that a free product of free groups amalgamated over a cyclic subgroup is LERF
Composition Algebras with Large Derivation Algebras

AbstractThe finite dimensional flexible composition algebras include the Hurwitz algebras (composition algebras with unit element), but also other interesting classes of algebras: the para-Hurwitz and the Okubo algebras. The above mentioned algebras present many symmetries, and this is reflected in their large derivation algebras. In the present paper we study the opposite question: What can be said about the composition algebras if we have some information about their derivation algebras? Our main result is the classification of all the composition algebras with such large derivation algebras