http://dspace.nbuv.gov.ua:80/handle/123456789/150357 2026-04-15T06:08:38Z http://dspace.nbuv.gov.ua:80/handle/123456789/154573 On modules over group rings of locally soluble groups for a ring of p -adic integers Dashkova, O.Yu. The author studies the Zp∞G-module A such that Zp∞ is a ring of p-adic integers, a group G is locally soluble, the quotient module A/CA(G) is not Artinian Zp∞-module, and the system of all subgroups H≤G for which the quotient modules A/CA(H) are not Artinian Zp∞-modules satisfies the minimal condition on subgroups. It is proved that the group G under consideration is soluble and some its properties are obtained. 2009-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/153385 Tiled orders of width 3 Zhuravlev, V.; Zhuravlyov, D. We consider projective cover over tiled order and calculate the kernel of epimorphism from direct sum of submodules of distributive module to their sum. 2009-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/153384 Algebra in the Stone-Čech compactification: applications to topologies on groups Protasov, I.V. For every discrete group G, the Stone-Čech compactification βG of G has a natural structure of compact right topological semigroup. Assume that G is endowed with some left invariant topology I and let τ¯ be the set of all ultrafilters on G converging to the unit of G in I. Then τ¯ is a closed subsemigroup of βG. We survey the results clarifying the interplays between the algebraic properties of τ¯ and the topological properties of (G,I) and apply these results to solve some open problems in the topological group theory. The paper consists of 13 sections: Filters on groups, Semigroup of ultrafilters, Ideals, Idempotents, Equations, Continuity in βG and G∗, Ramsey-like ultrafilters, Maximality, Refinements, Resolvability, Potential compactness and ultraranks, Selected open questions. 2009-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/153383 On action of outer derivations on nilpotent ideals of Lie algebras Maksimenko, D.V. Action of outer derivations on nilpotent ideals of Lie algebras are considered. It is shown that for a nilpotent ideal I of a Lie algebra L over a field F the ideal I+D(I) is nilpotent, provided that charF=0 or I nilpotent of nilpotency class less than p−1, where p=charF. In particular, the sum N(L) of all nilpotent ideals of a Lie algebra L is a characteristic ideal, if charF=0 or N(L) is nilpotent of class less than p−1, where p=charF. 2009-01-01T00:00:00Z