Group algebras of Lie nilpotency index 12 and 13

Let G be a group and let K be a field of characteristic p>0. Lie nilpotent group algebras of strong Lie nilpotency index up to 11 have already been classified. In this paper, our aim is to classify the group algebras KG which are strongly Lie nilpotent of index 12 or 13.     

The radical of a module category of string algebra

We study how to read from the ordinary quiver of a representation-finite string algebra, the minimal lower bound m ≥ 1 such that the mth power of the radical of its module category vanishes. We also show how to read the degree of any irreducible morphism in such a representation-finite string algebras considering strings.     

On the stable module category of a self-injective algebra

Let be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable -modules which belong to Auslander-Reiten components of the form or . The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.     

On Composition series relative to a module

ABSTRACT

A commutative algebra with the identity (a * b) * (c * d) ? (a * d) * (c * b) = (a, b, c) * d ? (a, d, c) * b is called Novikov–Jordan. Example: K[x] under multiplication a * b = ?(ab) is Novikov–Jordan. A special identity for Novikov–Jordan algebras of degree 5 is constructed. Free Novikov–Jordan algebras with q generators are exceptional for any q ≥ 1.

     

On a category of module bundles

It is shown, that the category ℰ _A (X) ofA-bundles over a topological spaceX is an additive pseudo-Abelian category for each Waelbroeck algebraA. This research was supported by Greek State Scholarship Foundation and Estonian Science Foundation grant 6205.     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.     

On the relationship between the length of an algebra and the index of nilpotency of its Jacobson radical

A sharp upper bound for the length of an algebra as a function of the index of nilpotency of its Jacobson radical and the length of the quotient algebra by the radical is obtained.     

On the efficiency index of one-point iterative processes

In this paper, we analyze the index of efficiency of one-point iterative processes, which are in practice the most used methods to solve a nonlinear equation. We obtain the best situation for one-point iterative processes with cubic convergence: Chebyshev’s method, Halley’s method, the super-Halley method and many others classical iterative methods with order of convergence three. By means of a construction of particular multipoint iterations, we get to improve the best situation obtained for one-point methods. Moreover, these type of multipoint iterations, can be considered as quasi-one-point iterations, since they only depend on one initial approximation. Numerical examples are given and the computed results support this theory. Partly supported by the Ministry of Education and Science (MTM 2005-03091) and the University of La Rioja (ATUR-05/43).     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

On the Classification of Commutative Right-Nilalgebras of Nilindex 5 and Dimension 4

Let D be a division ring with center k, and let D ^? be its multiplicative group. We investigate the existence of free groups in D ^?, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D ^?.     

The category of a map and the grade of a module

Letf: Y → X be a continuous map between connectedCW complexes. The homologyH _* (F) of the homotopy fibre is then a module over the loop space homologyH _* (ΩX). Theorem:If H _* (F; R) and H _* (ΩX; R) are R-free (R a principal ideal domain) then for some H _* (ΩX; R)-projective module P=P_>0 and for some m ≤ cat f: Some applications are also given. Research partially supported by a NATO travel grant. Research partially supported by an NSERC operating grant. U.A. au CNRS 751.     

On nilpotency of the barideal of a bernstein algebra

It is given a new bound for the index of nilpotency of the barideal of a finited generated Bernstein algebra. Also, it is found the index of solvability of the barideal of a Bernstein algebra and a bound of nilpotency for the square of the barideal.     

When the heart of a faithful torsion pair is a module category

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heartH(X,Y)of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.     

On the (co)homology theory of index category

In the present work the different types of regular index categories are given. The related cohomology groups of some of these categories are studied. We give some properties of index category related with monoid and algebra over it.     

On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups

Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this article, for a finite group G, we define the subgroup S(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups.