CENTRALIZERS AND INDUCTION

2007 ◽  
Vol 06 (03) ◽  
pp. 505-526 ◽  
Author(s):  
LARS KADISON
Keyword(s):  
Endomorphism Ring ◽  
Ring Homomorphism ◽  
Normal Subgroups ◽  
Ring Extension ◽  
Azumaya Algebra ◽  
Equivalence Of Categories ◽  
Tensor Square ◽  
Special Case

Given a ring homomorphism B → A, consider its centralizer R = AB, bimodule endomorphism ring S = End BAB and sub-tensor-square ring T = (A ⊗ BA)B. Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A | B is separable, H-separable, split or left depth two (D2). If RT or SR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which takes the place of the key module AAe for an Azumaya algebra A. In addition, we characterize left D2 extensions in terms of the module TR, and show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as pre-braided commutative. For example, the notion of normality yields a version for Hopf subalgebras of the fact that normal subgroups have normal centralizers, and yields a special case of a conjecture that D2 Hopf subalgebras are normal.

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

Spotting infinite groups

1999 ◽  
Vol 125 (1) ◽  
pp. 39-42 ◽  
Author(s):  
DANIEL ALLCOCK
Keyword(s):  
Lower Bounds ◽  
Normal Subgroups ◽  
Infinite Groups ◽  
Finitely Presented Group ◽  
Special Case ◽  
Finitely Presented

We generalize a theorem of R. Thomas, which sometimes allows one to tell by inspection that a finitely presented group G is infinite. Groups to which his theorem applies have presentations with not too many more relators than generators, with at least some of the relators being proper powers. Our generalization provides lower bounds for the ranks of the abelianizations of certain normal subgroups of G in terms of their indices. We derive Thomas's theorem as a special case.

The Homological Functors of Some Abelian Groups of Prime Power Order

2014 ◽  
Vol 71 (5) ◽  
Author(s):  
Rosita Zainal ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Samad Rashid
Keyword(s):  
Tensor Product ◽  
Prime Power ◽  
Homotopy Theory ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Schur Multiplier ◽  
Integer Coefficients ◽  
Nonabelian Tensor ◽  
Tensor Square ◽  
Special Case

The homological functors of a group were first introduced in homotopy theory. Some of the homological functors including the nonabelian tensor square and the Schur multiplier of abelian groups of prime power order are determined in this paper. The nonabelian tensor square of a group G introduced by Brown and Loday in 1987 is a special case of the nonabelian tensor product. Meanwhile, the Schur multiplier of G is the second cohomology with integer coefficients is named after Issai Schur. The aims of this paper are to determine the nonabelian tensor square and the Schur multiplier of abelian groups of order p5, where p is an odd prime

MARTINGALE CHARACTERIZATIONS OF INCREMENT PROCESSES IN A LOCALLY COMPACT GROUP

2003 ◽  
Vol 06 (04) ◽  
pp. 563-595 ◽  
Author(s):  
HERBERT HEYER ◽  
GYULA PAP
Keyword(s):  
Compact Group ◽  
Martingale Problem ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Fourier Theory ◽  
Group A ◽  
Tensor Square ◽  
Special Case

Martingale characterizations and the related martingale problem are studied for processes with independent (not necessarily stationary) increments in an arbitrary locally compact group. In the special case of a compact Lie group, a Lévy-type characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes.

scholarly journals On the non-albelian tensor square of a nilpotent group of class two

1994 ◽  
Vol 36 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Michael R. Bacon
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Homotopy Theory ◽  
Nonabelian Tensor ◽  
Image Position ◽  
Tensor Square ◽  
Special Case

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations,for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].

scholarly journals On free ring extensions of degreen

1981 ◽  
Vol 4 (4) ◽  
pp. 703-709
Author(s):  
George Szeto
Keyword(s):  
Galois Extension ◽  
Quaternion Algebra ◽  
Ring Extension ◽  
Quaternion Algebras ◽  
Free Ring ◽  
Azumaya Algebra ◽  
Ring Extensions ◽  
One To One ◽  
Generalized Quaternion

Nagahara and Kishimoto [1] studied free ring extensionsB(x)of degreenfor some integernover a ringBwith 1, wherexn=b,cx=xρ(c)for allcand somebinB(ρ=automophism of  B), and{1,x…,xn−1}is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternion algebras in whichb=−1andρis of order 2. The purpose of the present paper is to generalize a characterization of a generalized quaternion algebra to a free ring extension of degreenin terms of the Azumaya algebra. Also, it is shown that a one-to-one correspondence between the set of invariant ideals ofBunderρand the set of ideals ofB(x)leads to a relation of the Galois extensionBover an invariant subring underρto the center ofB.

Characterizations and Direct Sums of Unit-Endoregular Modules

2018 ◽  
Vol 61 (4) ◽  
pp. 1103-1112
Author(s):  
Xiaoxiang Zhang ◽  
Gangyong Lee
Keyword(s):  
Abelian Group ◽  
Endomorphism Ring ◽  
Regular Ring ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Baer Ring ◽  
Special Case

AbstractA module is called unit-endoregular if its endomorphism ring is unit-regular. In this paper, we continue the research in unit-endoregular modules. More characterizations of unit-endoregular modules are obtained. As a special case, we show that for an abelian group G, Endℤ(G) is a unit-regular Baer ring if and only if Endℤ(G) is a two-sided extending regular ring. While the class of unit-endoregular modules is not closed under direct sums, we provide a characterization when there are direct sums of two or more unit-endoregular modules also unit-endoregular under certain conditions. In particular, we investigate unit-endoregular modules which are direct sums of indecomposable modules.

A structure theorem on compact groups

2001 ◽  
Vol 130 (3) ◽  
pp. 409-426 ◽  
Author(s):  
KARL H. HOFMANN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Quotient Group ◽  
Structure Theorem ◽  
Cartesian Product ◽  
Simple Groups ◽  
Compact Groups ◽  
Finite Simple Groups ◽  
Normal Subgroups ◽  
Identity Component ◽  
Special Case ◽  
Totally Disconnected

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

2018 ◽  
Vol 41 ◽  
Author(s):  
Daniel Crimston ◽  
Matthew J. Hornsey
Keyword(s):  
General Theory ◽  
Biological Markers ◽  
Natural Environment ◽  
Relevant Dimension ◽  
Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

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