Publications (31)13.94 Total impact

Article: The Seriality of the Group Ring of a Finite Group Depends Only on the Characteristic of the Field
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ABSTRACT: It is proved that the seriality of the group ring of a finite group over a field depends only on the characteristic of the field.  [Show abstract] [Hide abstract]
ABSTRACT: We will develop the model theory of modules over commutative Bezout domains. In particular we characterize commutative Bezout domains B whose lattice of ppformulae has no width and give some applications to the existence of superdecomposable pure injective Bmodules.  [Show abstract] [Hide abstract]
ABSTRACT: For a given p, we determine when the pmodular group ring of a group from GL(n, q), SL(n, q) and PSL(n, q)series is serial.  [Show abstract] [Hide abstract]
ABSTRACT: We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.  [Show abstract] [Hide abstract]
ABSTRACT: We classify indecomposable pure injective modules over 1domestic string algebras verifying Ringel’s conjecture on their structure.  [Show abstract] [Hide abstract]
ABSTRACT: We describe the Ziegler spectrum of a Bézout domain B = D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of Bmodules when D is “sufficiently” recursive.  [Show abstract] [Hide abstract]
ABSTRACT: Let R be an niterated ring of differential polynomials over a commutative noetherian domain which is a ℚalgebra. We will prove that for every proper ideal I of R, the (n + 1)iterated intersection I(n + 1) of powers of I equals zero. A standard application includes the freeness of nonfinitely generated projective modules over such rings. If I is a proper ideal of the universal enveloping algebra of a finitedimensional solvable Lie algebra over a field of characteristic zero, then we will improve the above estimate by showing that I(2) = 0.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that every pure projective torsion free module over a (commutative) Bass domain with finite normalization contains a finitely generated direct summand.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that for any finite psolvable group G with a cyclic pSylow subgroup and any field F of characteristic p, the group ring FG is serial. As a corollary for an arbitrary field we will produce a list of all groups of order ≤ 100 whose group rings are serial.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that the KrullGabriel dimension of the category of modules over any 1domestic nondegenerate string algebra is 3.  [Show abstract] [Hide abstract]
ABSTRACT: We classify all (finitely generated or not) projective modules over a class of semilocal ring constructed using nealy simple uniserial domains. They in turn are connected with noncommutative valuations constructed using embeddings of right ordered groups into skew fields.  [Show abstract] [Hide abstract]
ABSTRACT: We will describe the torsionfree part of the Ziegler spectrum, both the points and the topology, over the integral group ring of the Klein group. For instance we will show that the Cantor–Bendixson rank of this space is equal to 3.  [Show abstract] [Hide abstract]
ABSTRACT: We develop a general strategy of classifying generalized lattices over orders of finite lattice type and demonstrate the effectiveness of this approach on various examples.  [Show abstract] [Hide abstract]
ABSTRACT: We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules.  [Show abstract] [Hide abstract]
ABSTRACT: We classify group rings of finite groups over a field F according to the modeltheoretic complexity of the category of their modules. For instance, we prove that, if F contains a primitive cubic root of 1, then the Krull–Gabriel dimension of such rings is 0, 2, or undefined. 
Article: How to construct a ‘concrete’ superdecomposable pureinjective module over a string algebra
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ABSTRACT: We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an example, let R be the endomorphism ring of a countable dimensional vector space V, and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R ′ = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ′ ∼ = eR ′ ⊕ (1 − e)R ′ as a right module over itself and R ′ ∼ = eR ′ ∼ = (1 − e ′)R ′. Furthermore, every nontrivial decomposition of R ′ as a right module is of this form, therefore R ′ is superdecomposable. For more examples, let R = k〈X, Y 〉 be a free algebra over a field k, and let E = E(RR) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that RR has no nonzero uniform submodules. Therefore the same is true for E, hence E is a superdecomposable injective module. More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pureinjective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pureinjective if M is a direct summand of a direct product of finite dimensional Amodules. If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pureinjective Amodule has an indecomposable direct summand. But recently Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pureinjective module (note that every string algebra is tame). The main  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the structure of the socalled Gerasimov–Sakhaev counterexample, which is a particular example of a universal localization, and classify (both finitely and infinitely generated) projective modules over it.  [Show abstract] [Hide abstract]
ABSTRACT: We characterize rings over which every projective module is a direct sum of finitely generated modules, and give various examples of rings with and without this property.  [Show abstract] [Hide abstract]
ABSTRACT: We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author (J. Pure Appl. Algebra 208(2), 2007). We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular and therefore does not satisfy the strong form of the generating hypothesis.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable.
Publication Stats
135  Citations  
13.94  Total Impact Points  
Top Journals
Institutions

20122015

Belarusian State University
 Department of Mathematics
Myenyesk, Minsk, Belarus


20102011

Plekhanov Russian Academy of Economics
Moskva, Moscow, Russia


20032011

The University of Manchester
 School of Mathematics
Manchester, England, United Kingdom


20032005

The Ohio State University
 Department of Mathematics
Columbus, Ohio, United States
