Morita equivalence for infinite matrix rings

In this paper, weak distinguished subcategory and distinguished subcategory of modules are introduced. Left(right) local unital rings are particularly considered. Also, representable equivalent functors between categories. By using the replacement techniques of modules, a general theory of Morita equivalence for infinite matrix rings is established. This theory not only extends the classical Morita theory of equivalence from finite matrix rings to infinite matrix rings and also contains some new results which are useful in studying the algebraic structures for infinite matrix rings. Some results of classical Morita theory are included as its special cases.     

Isomorphisms of graded endomorphism rings of graded modules close to free ones

We obtain criteria that answer the question of when an isomorphism of graded endomorphism rings of strong gr-generators is induced by a gr-generator, graded Morita equivalence, or semi-linear isomorphism. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 3–18, 2007.     

McCoy rings and zero-divisors

We investigate relations between the McCoy property and other standard ring theoretic properties. For example, we prove that the McCoy property does not pass to power series rings. We also classify how the McCoy property behaves under direct products and direct sums. We prove that McCoy rings with 1 are Dedekind finite, but not necessarily Abelian. In the other direction, we prove that duo rings, and many semi-commutative rings, are McCoy. Degree variations are defined, studied, and classified. The McCoy property is shown to behave poorly with respect to Morita equivalence and (infinite) matrix constructions.     

Stable equivalences of Morita type and stable Hochschild cohomology rings

We prove that a stable equivalence of Morita type between finite dimensional algebras preserves the stable Hochschild cohomology rings, that is, Hochschild cohomology rings modulo the projective center, thus generalizing the results of Pogorzały and Xi.     

Isomorphisms of formal matrix incidence rings

Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.     

Anti-isomorphisms of graded endomorphism rings of graded modules close to free ones

We obtain criteria that answer the question of when an anti-isomorphism of graded endomorphism rings of the strict gr-generators is induced by a graded Morita anti-equivalence or a graded anti-semilinear isomorphism.     

Isomorphisms of graded endomorphism rings of progenerators

We prove the analogue of Bolla’s theorem that isomorphisms of graded endomorphism rings of progenerators are induced by the graded Morita equivalence __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 3–10, 2007.     

The cancellation property of rings

A ring R is said to have the finitely generated cancellation property provided that the module isomorphism R⊕B?R⊕C implies B?C for any finitely generated R-modules B and C. It is proved that R has this property is equivalent to the existence of the cancellation matrices over R. Moreover, the structure of such matrices is investigated and finite weakly stable rings are characterized in terms of their cancellation matrices.     

Related comparability over exchange rings

It is shown that every exchange ring satisfying related comparability is separative. This yields that related comparability over exchange rings is Morita invariant. Also we investigate pseudosimilarity over exchange rings satisfying related comparability.     

Stable equivalence and rings whose modules are a direct sum of finitely generated modules

The representation theory of a ring Δ has been studied by examining the category of contravariant (additive) functors from the category of finitely generated left Δ-modules to the category of abelian groups. Closely connected with the representation theory of a ring is the study of stable equivalence, which is a relaxing of the notion of Morita equivalence. Here we relate two stably equivalent rings via their respective functor categories and examine left artinian rings with the property that every left Δ-module is a direct sum of finitely generated modules.     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

Primitive involution rings

Summary A *-primitive involution ring Ris either a left and right primitive ring or a certain subdirect sum of a left primitive and a right primitive ring with involution exchanging the components. An example is given of a left and right primitive ring which admits no row and column finite matrix representation. We characterize *-primitive involution rings in terms of maximal *-biideals. A *-prime involution ring has a minimal left ideal if and only if it has a minimal *-biideal, and these involution rings are always *-primitive.     

Local rings of exchange rings

We characterize the exchange property for non-unital rings in terms of their local rings at elements,and we use this characterization to show that the exchange property is Morita invariant for idempotent rings.We also prove that every ring contains a greatest exchange idela(with respect to the inclusion).     

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz–Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Δ-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.     

Waring's problem in finite rings

In this paper we obtain explicit results for Waring's problem over general finite rings, especially matrix rings over finite fields by building on analogous results over finite fields. Commutative algebra, in particular the Jacobson radical and nilpotent ideals, plays an important role in our proofs.     

Parametrization of central Frattini extensions and isomorphisms of small group rings

We solve the modular isomorphism problem for small group rings, i.e., we determine, for a given finite p-group H, precisely which central Frattini extensions of H give rise to isomorphic small group rings over the field with p elements. The first author acknowledges support by the Deutsche Forschungsgemeinschaft.     

余代数和Morita-Takeuchi关系

本文证明了若余代数Ｃ和Ｄ是Ｍｏｒｉｔａ Ｔａｋｅｕｃｈｉ等价的，则Ｃ的子余代数格和Ｄ的子余代数格同构．设ＭΓ，ＮΓ是拟有限右Γ 余模，则（ｈ－Γ（Ｍ，Ｍ），ｈ－Γ（Ｎ，Ｎ）；ｈ－Γ（Ｎ，Ｍ），ｈ－Γ（Ｍ，Ｎ）；ｆ，ｇ）有Ｍｏｒｉｔａ Ｔａｋｅｕｃｈｉ关系．并给出了此Ｍ Ｔ关系是Ｍ Ｔ等价条件的及由已知的Ｍ Ｔ关系构造新的Ｍ Ｔ关系的方法．     

Mauldin-Williams graphs, Morita equivalence and isomorphisms

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras.

We also investigate the relationship between the Morita equivalence and isomorphism class of the -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

     

Incidence Category of the Young Lattice,Injections Between Finite Sets,and Koszulity

We study the quadratic quotients of the incidence category of the Young lattice defined by the zero relations corresponding to adding two boxes to the same row,or to the same column,or both.We show that the last quotient corresponds to the Koszul dual of the original incidence category,while the first two quotients are,in a natural way,Koszul duals of each other and hence they are in particular Koszul self-dual.Both of these two quotients are known to be basic representatives in the Morita equivalence class of the category of injections between finite sets.We also present a new,rather direct,argument establishing this Morita equivalence.