A Uniqueness Theorem in a Finitely Accessible Additive Category

It is proved that if A ₁,A ₂,..., A _m and B ₁,B ₂,..., B _n are objects in a finitely accessible additive category \(\mathcal{A}\) such that their pure injective envelopes are indecomposable, and there are pure monomorphisms μ:A ₁?⊕?A ₂?⊕?...?⊕?A _m →B ₁?⊕?B ₂?⊕?...?⊕?B _n and ν:B ₁?⊕?B ₂?⊕?...?⊕?B _n →A ₁?⊕?A ₂?⊕?...?⊕?A _m , then m?=?n and there are a permutation σ and pure monomorphisms A _i →B _σ(i) and B _σ(i)→A _i for every i?=?1, 2, ..., n.     

关于半局部群环

Let R be an associative ring with identity.R is said to be semilocal if R/J(R)is(semisimple)Artinian,where J(R)denotes the Jacobson radical of R.In this paper,we give necessary and sufficient conditions for the group ring RG to be semilocal,where G is a locally finite nilpotent group.     

On QB Endomorphism Rings

We study, in this article, QB endomorphism rings of quasi-projective modules by means of completions of diagrams. The dual problems for quasi-injective modules are also investigated.     

Local Morphisms and Modules with a Semilocal Endomorphism Ring

We study local morphisms in the setting of general noncommutative rings. In particular, we apply local morphisms to study endomorphism rings of modules. We use our constructions to determine classes of modules with semilocal endomorphism rings. For instance, we prove that every finitely presented right module over a semilocal ring has a semilocal endomorphism ring.Alberto Facchini was partially supported by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica, Italy, and by Università di Padova (Progetto di Ateneo CDPA048343 “Decomposition and tilting theory in modules, derived and cluster categories”).Dolors Herbera was partially supported by the DGI and the European Regional Development Fund, jointly, through Project BFM2002-01390, and by the Comissionat per Universitats i Recerca of the Generalitat de Catalunya.     

On Modules over Endomorphism Algebras of Maximal Rigid Objects in 2-Calabi-Yau Triangulated Categories

This note investigates the modules over the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated categories. We study the possible complements for almost complete tilting modules. Combining with Happel's theorem, we show that the possible exchange sequences for tilting modules over such algebras are induced by the exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yau triangulated categories. For the modules of infinite projective dimension, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.     

关于正则模的自同态环

本文研究正则模Ｐ的自同态环Ｓ，主要给出使Ｓ的稳定秩为１的Ｐ的几个充分条件．     

Commutative Semilocal Rings Whose Additive or Multiplicative Groups Have Finite Rank

ABSTRACT

Let R be an infinite semilocal ring. Then R is not finitely generated, neither the additive group R(+), nor the multiplicative group R? (of invertible elements) is minimax and at least one of these groups has infinite Prüfer rank.     

On the Additive Property of Finitely Additive Measures

Journal of Theoretical Probability - By additive property, we refer to a condition under which $$L^p$$ spaces over finitely additive measures are complete. In their 2000 paper, Basile and Rao gave...     

Regularity in Endomorphism Rings

The unique largest regular ideal Reg(A, A) in the endomorphism ring End(A) is computed for abelian groups A using the general tools developed in [7 Kasch , F. , Mader , A. ( 2005 ). Regularity and substructures of hom . Communications in Algebra 34 : 1459 – 1478 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. This generalizes earlier results on groups with regular endomorphism ring. Interesting questions remain for a very special class of mixed abelian groups.     

On Finitely Annihilated Modules and Artinian Rings

Let R be a ring. An R-module M is finitely annihilated if the annihilator of M is the annihilator of a finite subset of M. It is proved that if R has right socle S then the ring R/S is right Artinian if and only if every singular right R-module is finitely annihilated. Moreover, a right Noetherian ring R is right Artinian if and only if every uniform right R-module is finitely annihilated. In addition, a (right and left) Noetherian ring is (right and left) Artinian if and only if every injective right R-module is finitely annihilated. This paper will form part of the Ph.D. thesis at the University of Glasgow of the second author. He would like to thank the EPSRC for their financial support     

On Integral Fusion Categories with Low-Dimensional Simple Objects

Let k be an algebraically closed field of characteristic zero. In this article, we consider an integral fusion category over k in which the Frobenius–Perron dimensions of its simple objects are at most 3. We prove that such fusion category is of Frobenius type. In addition, we also prove that such fusion category is not simple.     

On Rings for Which Finitely Generated Ideals Have Only Finitely Many Minimal Components

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R are always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over R as well as to R-algebras which are finitely presented as R-modules.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Abelian Groups with Annihilator Ideals of Endomorphism Rings

We describe the periodic groups whose endomorphism rings satisfy the annihilator condition for the principal left ideals generated by nilpotent elements. We prove that torsion-free reduced separable, vector, and algebraically compact groups have endomorphism rings with the annihilator condition for the principal left (right) ideals generated by nilpotent elements if and only if these rings are commutative. We show that the almost injective groups (in the sense of Harada) are injective, i.e. divisible.     

Completely Decomposable Abelian Groups with Clean Endomorphism Rings

Conditions for cleanness of endomorphism rings of completely decomposable Abelian groups are established.     

A Continually Descending Endomorphism of a Finitely Generated Residually Finite Group

Let :GG be an endomorphism of a finitely generated residuallyfinite group. R. Hirshon asked if there exists n such that therestriction of to ⁿ(G) is injective. We give an example toshow that this is not always the case. 1991 Mathematics SubjectClassification 20E26.     

On Additive Commutator Groups in Division Rings

Let D be a division ring with center F and denote by [D,D] the group generated additively by additive commutators. First, it is shown that in zero characteristic, D is algebraic over F if and only if each element of [D,D] is algebraic over F. We conjecture that this assertion is true for any characteristic. Also, as a generalization of Jacobson’s Theorem it is proved that D is an F-central division ring if and only if all its additive commutators are of bounded degree over F. Furthermore, we study the F- vector space D/[D,D] and show that dim_FD/[D,D] ≤ 1 if D is algebraic over F and char F = 0. We then prove that any algebraic division ring contains a separable additive commutator over F except in one special case. Finally, the existence of primitive elements in [D, D] is studied for finite separable extensions of F in D.     

On Additive Maps and Commutativity in Rings

We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Bre?ar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) ? d(y)d(x) = 0 for all x, y in a suitably chosen subset of R.     

Automorphic Objects in Categories

Siberian Mathematical Journal -