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1.
The base radical class L
b( X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, L
b( X) is the lower radical class defined by X, but otherwise X may not be contained in L
b( X). We prove that for a hereditary radical class L with semisimple class S( R), L
b( S( R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form L
b( X) are discussed.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
2.
We consider the disjunction property, DP, in the class of extensions of minimal logic L
j
. Conditions are described under which DP is translated from the class PAR of properly paraconsistent extensions of the logics of class L
j
into the class INT of intermediate extensions and the class NEG of negative extensions, and conditions for its being translated back into PAR. The logic L
F
in PAR, which specifies conditions for DP to be translated from PAR into NEG, is defined and is characterized in terms of j-algebras and Kripke frames. Moreover, we show that L
F
is decidable and possesses the disjunction property. 相似文献
3.
The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C′ are linearly equivalent if there exists a non-singular matrix A such that AC = C′ where C and C′ are matrices with the code words of C and C′ as columns. Hessler proved that the perfect codes C and C′ are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe
all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel
of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side
class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient
to find the family of all kernels of perfect codes. 相似文献
5.
ABSTRACT Let K denote a commutative ring with unity and A be a K-algebra. An element, d ∈ A is said to be left self distributive, or LSD, if dxy = dx dy for all x, y ∈ A. Let ?( A) be the set of LSD elements. Similarly, one can define the set of right self distributive, or RSD, elements and let ?( A) be the set of RSD elements. Let 𝒟( A) = ?( A) ∩ ?( A), the set of self distributive, or SD, elements. An algebra, A, is said to be left self distributively generated, or LSD-generated, if A = mod K (?( A)), the K- module generated by ?( A). Analogously, one defines RSD-generated and SD-generated algebras. If A = mod K (?( A)) = mod K (?( A)), then A is said to be LSD/RSD-generated, which is a strictly larger class than the class of SD-generated algebras. Examples are given to illustrate the variety of LSD-generated algebras. This paper continues the study of LSD-generated, RSD-generated, LSD/RSD-generated and SD-generated algebras. This paper characterizes exactly which structural matrix rings are LSD-generated. The paper begins with an important lemma that characterizes LSD elements in a matrix ring in terms of the entries of the matrix. The main result characterizes those structural matrix rings that are LSD-generated, first in terms of a 2 × 2 generalized matrix ring, then strictly in terms of the shape of the matrix ring. Sharper results are obtained for LSD/RSD-generated and SD-generated structural matrix rings. The final section is devoted to an application of this result to endomorphism rings. If the endomorphism ring of a finitely generated module is a homomorphic image of a structural matrix ring, then the module is a direct sum of cyclic modules. Further conditions are given to describe when the structural matrix ring is LSD-generated, in terms of the annihilators of the generating set. 相似文献
6.
Abstract Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U n ( R) be the subgroup of GL( V) that preserves ( , ). Let SU n ( R) be the subgroup of all g ∈ U n ( R) whose determinant is equal to one. Let Ψ be the Weil character of U n ( R). All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU n ( R). 相似文献
7.
ABSTRACT If G is a nontrivial finite group with the property that χ ( 1) 2 divides bi| G | for all irreducible (complex) characters χ , then G has a nontrivial normal abelian subgroup. 相似文献
8.
ABSTRACT We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006
Faith , C. ( 2006 ). Factor rings of pseudo-Frobenius rings . J. Algebra and Its Applications 6 :(to appear). [CSA] [Web of Science ®] , [Google Scholar]). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5). These results also generalize results of the author in Faith ( 1985
Faith , C. ( 1985 ). The maximal regular ideal of self-injective and continuous rings splits off . Arch. Math. 44 : 511 – 521 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring. The results are applied in Section 6 to extend theorems of Faith ( 1996
Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) characterizing VNR rings, and, as the title of Faith ( 1996
Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) suggests, extend the conjecture of Shamsuddin. 相似文献
9.
The class of fully copositive ( C
0
f
) matrices introduced in [G.S.R. Murthy, T. Parthasarathy, SIAM Journal on Matrix Analysis and Applications 16 (4) (1995) 1268–1286] is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class of Q
0-matrices are P
0-matrices. As a corollary of this main result, we establish that a bisymmetric Q
0-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization of Q
0-matrices within the class of C
0
f
. While establishing this characterization, it will be shown that Graves's principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable to C
0
f
Q
0 class. As a byproduct of this characterization, we observe that a C
0
f
-matrix is in Q
0 if, and only if, it is completely Q
0. Also, from Aganagic and Cottle's [M. Aganagic, R.W. Cottle, Mathematical Programming 37 (1987) 223–231] result, it is observed that LCPs arising from C
0
f
Q
0 class can be processed by Lemke's algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author. 相似文献
10.
First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex
euclidean ball B⊂ C
n
with its relative logarithmic capacity in C
n
with respect to the same ball B. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of C
n
is also proved.
Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative
size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities
at infinity on C
n
as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂( C
n
.
Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic
functions.
This work was partially supported by the programmes PARS MI 07 and AI.MA 180. 相似文献
11.
AbstractGiven a topological space X = ( X, T ), we show in the Zermelo-Fraenkel set theory ZF that: Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T4 and the countable axiom of choice holds true. We also show:It is relatively consistent with ZF the existence of a non countably compact T1 space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite. It is relatively consistent with ZF the existence of a countably compact T4 space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets. 相似文献
13.
Various problems are considered in an attempt to generalize the simplex algorithm of linear programming to a much wider class of convex bodies than the class of convex polytopes. A conjecture of D.G. Larman and C.A. Rogers is disproved by constructing a three-dimensional convex body K with an extreme point e, so that for a certain linear functional f, there are no paths in the one-skeleton of K leading from e, along which f strictly increases. Their conjectured generalization is, however, proved for the large class of three-dimensional convex bodies, all of whose extreme points are exposed.A strong generalization of the simplex algorithm is obtained for the class of all finite-dimensional convex bodies, where, for a given exposed point e of a convex body K, it is possible to find f-strictly-increasing paths in the one-skeleton of K, leading from e, for almost all linear functionals f.Research sponsored by the British Science Research Council. 相似文献
14.
We give a characterization of the class Co( F)\mathbf{Co}(\mathcal{F}) [ Co( Fn)\mathrm{\mathbf{Co}}(\mathcal{F}_n), n < ω, respectively] of lattices isomorphic to convexity lattices of posets which are forests [forests of length at most n, respectively], as well as of the class Co( L)\mathbf{Co}(\mathcal{L}) of lattices isomorphic to convexity lattices of linearly ordered posets. This characterization yields that the class of finite
members from Co( F)\mathbf{Co}(\mathcal{F}) [from Co( Fn)\mathbf{Co}(\mathcal{F}_n), n < ω, or from Co( L)\mathbf{Co}(\mathcal{L})] is finitely axiomatizable within the class of finite lattices. 相似文献
16.
Iterating the triple construction applied consecutively to n Boolean algebras, we introduce two finitely axiomatizable subclasses SAin{{\bf SA}^{\rm i}_n} and SAsn{{\bf SA}^{\rm s}_n} of the class SA
n
of all Stone algebras of degree n with all the structure homomorphisms in their P-product representation injective or surjective, respectively. Then the class
of all Post algebras of degree n is definitionally equivalent to the intersection SAin ? SAsn{{\bf SA}^{\rm i}_{n} \cap {\bf SA}^{\rm s}_{n}}. We show that for each n ≥ 2 the class SAin{{\bf SA}^{\rm i}_n} is hereditarily undecidable while SAsn{{\bf SA}^{\rm s}_{n}} is decidable. As a consequence we obtain several (un)decidability results for various axiomatic classes of Stone algebras:
among them the decidability of the class of all Stone algebras of degree n which are dually pseudocomplemented and form a dual Stone algebra under the operation of dual pseudocomplement, and undecidability
of the class of all Stone algebras with Boolean dense set. On the other hand, the class of all finite members in SA
n
is decidable. 相似文献
17.
When we interpret modal ? as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ? as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}When we interpret modal ◊ as the limit point operator of a topological space, the G?del-L?b modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S
α
, where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ◊ as closure in a topological
space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H
α
, where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H
α
in terms of α-representations. We prove that X ? H1{X \in {\bf H}_{1}} iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that X ? Hn{X \in {\bf H}_{n}} iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have Ha ?Scat = Sb+2n-1 èSb+2n{{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}} , and for a limit ordinal α, we have Ha ?Scat = Sa{{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}} . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. 相似文献
18.
This paper addresses Infinitesimal Perturbation Analysis ( IPA) in the class of Make-to Stock ( MTS) production-inventory systems with backorders under the continuous-review ( R, r) policy, where R is the stock-up-to level and r is the reorder point. A system from this class is traditionally modeled as a discrete system with discrete demand arrivals
at the inventory facility and discrete replenishment orders placed at the production facility. Here, however, we map an underlying
discrete MTS system to a Stochastic Fluid Model ( SFM) counterpart in which stochastic fluid-flow rate processes with piecewise constant sample paths replace the corresponding
traditional discrete demand arrival and replenishment stochastic processes, under very mild regularity assumptions. The paper
then analyzes the SFM counterpart and derives closed-form IPA derivative formulas of the time-averaged inventory level and
time-averaged backorder level with respect to the policy parameters, R and r, and shows them to be unbiased. The obtained formulas are comprehensive in the sense that they are computed for any initial
inventory state and any time horizon, and are simple and fast to compute. These properties hold the promise of utilizing IPA
derivatives as an ingredient of offline design algorithms and online management and control algorithms of the class of systems
under study.
相似文献
19.
Abstract Structure theorems are obtained for certain radical classes of rings (including the Brown-McCoy radical class, the class of λ-rings, the class of E 5-rings, the class of E 6-rings and the class of f-regular rings) by generalizing the concept of a prime ideal. 相似文献
20.
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [ 27 Petrogradsky, V. M. (2006). Examples of self-iterating Lie algebras. J. Algebra 302(2):881–886.[Crossref], [Web of Science ®] , [Google Scholar]], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [ 39 Shestakov, I. P., Zelmanov, E. (2008). Some examples of nil Lie algebras. J. Eur. Math. Soc. (JEMS) 10(2):391–398.[Crossref], [Web of Science ®] , [Google Scholar]]. There are a few more examples of self-similar finitely generated restricted Lie algebras with a nil p-mapping, but, as a rule, that algebras have no clear basis and require technical computations. Now we construct a family L( Ξ) of 2-generated restricted Lie algebras of slow polynomial growth with a nil p-mapping, where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. Namely, GKdimL( Ξ)≤2 for all such algebras. The algebras are constructed in terms of derivations of infinite divided power algebra Ω. We also study their associative hulls A? End(Ω). Algebras L and A are ?2-graded by a multidegree in the generators. If Ξ is periodic then L( Ξ) is self-similar. As a particular case, we construct a continuum subfamily of non-isomorphic nil restricted Lie algebras L( Ξα), α∈? +, with extremely slow growth. Namely, they have Gelfand-Kirillov dimension one but the growth is not linear. For this subfamily, the associative hulls A have Gelfand-Kirillov dimension two but the growth is not quadratic. The virtue of the present examples is that they have clear monomial bases. 相似文献
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