Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
Solutions of a linear equation b=ax in a homomorphic image of a commutative Bezout domain of stable range 1.5 is developed. It is proved that the set of solutions of a solvable linear equation contains at least one solution that divides the rest, whi
ch is called a generating solution. Generating solutions are pairwise associates. Using this result, the structure of elements of the Zelisko group is investigated.
For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exis
ts a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.
A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to
Prufer domains. In the present paper we investigate some analogs of these concepts for modules over group rings.
Let FL_s(K) be the finitary linear group of degree s over an associative ring K with unity. We prove that the torsion subgroups of FL_s(K) are locally finite for certain classes of rings K. A description of some f.g. solvable subgroups of FL_s(K) are given.
Let R[G] be the group ring of a group G over an associative ring R with unity such that all prime divisors of orders of elements of G are invertible in R. If R is finite and G is a Chernikov (torsion FC-) group, then each R-derivation of R[G] is inne
r. Similar results also are obtained for other classes of groups G and rings R.
We introduce the notion of a super-representation of a quiver. For super-representations of quivers over a field of characteristic zero, we describe the corresponding (super)algebras of polynomial semi-invariants and polynomial invariants.
We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout domains.
Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determin
ed by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.
All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.
