scholarly journals SUBSTRUCTURES OF HOM

2011 ◽  
Vol 10 (01) ◽  
pp. 119-127 ◽  
Author(s):  
TSIU-KWEN LEE ◽  
YIQIANG ZHOU
Keyword(s):  
Jacobson Radical ◽  
The Other ◽  
Natural Question ◽  
Singular Ideal

Let M and N be two modules over a ring R. The concern is about the four substructures of hom R(M, N): the Jacobson radical J[M, N], the singular ideal Δ[M, N], the co-singular ideal ∇[M, N] and the total Tot [M, N]. One natural question is to characterize when the total is equal to one or more of the other structures. We review some known results and prove several new results towards this question and, as consequences, give answers to some related questions.

scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

scholarly journals Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations

2020 ◽  
Vol 379 (3) ◽  
pp. 1077-1112 ◽  
Author(s):  
György Pál Gehér ◽  
Peter Šemrl
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Mathematical Structure ◽  
Quantum Measurements ◽  
The Other ◽  
Effect Algebras ◽  
Natural Question ◽  
Fuzzy Event ◽  
Space Effect ◽  
Symmetry Transformations

Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.

Triangle Taste Tests: Are the Subjects who Respond Correctly Lucky or Good?

1981 ◽  
Vol 45 (3) ◽  
pp. 111-119 ◽  
Author(s):  
Donald G. Morrison
Keyword(s):  
The Other ◽  
Natural Question ◽  
Weighted Voting ◽  
Taste Test ◽  
Preference Tests ◽  
Subsequent Preference

When a triangle taste test, i.e., “Which of these three stimuli is different from the other two?”, is given twice to each subject, a natural question to ask is: “Are the subjects who got both tests correct really better discriminators than the other subjects who got 0 or 1 correct?” Surprisingly enough, the answer to this question has nothing to do with the number of subjects who got both correct, i.e., the twos. Rather, it is the ratio of the ones to the zeroes that supplies the necessary information. This information on the subject's abilities to discriminate can be used to develop a weighted voting procedure for subsequent preference tests.

scholarly journals Is It Possible to Determine Whether a Star is Rotating About a Unique Axis?

1993 ◽  
Vol 137 ◽  
pp. 566-568 ◽  
Author(s):  
D.O. Gough ◽  
A.G. Kosovichev
Keyword(s):  
Simple Model ◽  
Angular Velocity ◽  
Rotation Axis ◽  
The Other ◽  
The Sun ◽  
Natural Question ◽  
Rotating Stars ◽  
The Core ◽  
Axis Of Rotation ◽  
Fixed Axis

Rotating stars are normally presumed to rotate about a unique axis. Would it be possible to determine whether or not that presumption is correct? This is a natural question to raise, particularly after the suggestion by T. Bai & P. Sturrock that the core of the sun rotates about an axis that is inclined to the axis of rotation of the envelope.A variation with radius of the direction of the rotation axis would modify the form of rotational splitting of oscillation eigenfrequencies. But so too does a variation with depth and latitude in the magnitude of the angular velocity. One type of variation can mimic the other, and so frequency information alone cannot differentiate between them. What is different, however, is the structure of the eigenfunctions. Therefore, in principle, one might hope to untangle the two phenomena using information about both the frequencies and the amplitudes of the oscillations.We consider a simple model of a star which is divided into two regions, each of which is rotating about a different fixed axis. We enquire whether there are any circumstances under which it might be possible to determine seismologically the separate orientations of the axes.

scholarly journals On the Dimension of Modules and Algebras, IV: Dimension of Residue Rings of Hereditary Rings

1956 ◽  
Vol 10 ◽  
pp. 87-95 ◽  
Author(s):  
Samuel Eilenberg ◽  
Hirosi Nagao ◽  
Tadasi Nakayama
Keyword(s):  
Left Ideal ◽  
Jacobson Radical ◽  
Unit Element ◽  
The Other ◽  
Dedekind Ring ◽  
Integral Domains ◽  
Hereditary Ring ◽  
Common Value ◽  
Residue Ring ◽  
The Common

A ring (with unit element) Λ is called semi-primary if it contains a nilpotent two-sided ideal N such that the residue ring Γ = Λ/N is semi-simple (i.e. l.gl.dim Γ = r.gl.dim Γ = 0). N is then the (Jacobson) radical of Λ. Auslander [1] has shown that if Λ is semi-primary thenThe common value is denoted by gl. dim Λ. On the other hand, for any ring Λ the following conditions are equivalent : (a) 1. gl. dim Λ ≦ 1, (b) each left ideal in Λ is projective, (c) every submodule of a projective left Λ-module is projective. Rings satisfying conditions (a)-(c) are called hereditary. For integral domains the notions of “hereditary ring” and “Dedekind ring” coincide.

scholarly journals Random-Player Maker-Breaker games

10.37236/5032 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Krivelevich ◽  
Gal Kronenberg
Keyword(s):  
Perfect Matching ◽  
Winning Strategy ◽  
Hamilton Cycle ◽  
The Other ◽  
Natural Question ◽  
Vertex Connectivity ◽  
Asymptotic Values ◽  
Player Game ◽  
Matching Game ◽  
As If

In a $(1:b)$ Maker-Breaker game, one of the central questions is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims $b$ (or $m$) elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the $(1:b)$ random-Breaker game and the $(m:1)$ random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the $k$-vertex-connectivity game (played on the edge set of $K_n$). For each of these games we find or estimate the asymptotic values of the bias (either $b$ or $m$) that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of the bias.

scholarly journals RADIKAL SUPERNILPOTENT BERTINGKAT

2019 ◽  
Vol 2 (1) ◽  
pp. 1
Author(s):  
Puguh Wahyu Prasetyo
Keyword(s):  
Jacobson Radical ◽  
Ring Theory ◽  
The Other ◽  
Graded Rings ◽  
Radical Theory ◽  
Other Hand

The development of Ring Theory motivates the existence of the development of the Radical Theory of Rings. This condition is motivated since there are rings which have properties other than those owned by the set ring of all integers. These rings are collected so that they fulfill certain properties and they are called radical classes of rings. As the development of science about how to separate the properties of radical classes of rings motivates the existence of supernilpotent radical classes. On the other hand, there exists the concept of graded rings. This concept can be generalized into the Radical Theory of Rings. Thus, the properties of the graded supernilpotent radical classes are very interesting to investigate. In this paper, some graded supernilpotent radical of rings are given and their construction will be described. It follows from this construction that the graded Jacobson radical is a graded supernilpotent radical.

Representation of tiled matrix rings as full matrix rings

1989 ◽  
Vol 105 (1) ◽  
pp. 67-72 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Positive Integer ◽  
Jacobson Radical ◽  
Integral Domain ◽  
Matrix Ring ◽  
The Other ◽  
Matrix Rings ◽  
Full Matrix ◽  
Image Position ◽  
Full Matrix Ring ◽  
Unique Maximal Ideal

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

scholarly journals When does a semiring become a residuated lattice?

2016 ◽  
Vol 09 (04) ◽  
pp. 1650088
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
The Other ◽  
Natural Question ◽  
Double Negation ◽  
Complete Distributivity ◽  
Completely Distributive ◽  
Double Negation Law ◽  
The Given ◽  
Converse Assertion

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

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