A Note on Extensions of Bear and P.p.-rings
We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.
Suppose is a prime number and is a commutative ring with unity of characteristic 0 in which is not a unit. Assume that and are -primary abelian groups such that the respective group algebras and are -isomorphic. Under certain restrictions on the ideal structure of , it is shown that and are isomorphic.
It is proved that if G is a locally (soluble-by-finite) group of infinite rank in which every proper subgroup of infinite rank contains an abelian subgroup of finite index, then all proper subgroups of G are abelian-by-finite.
The structure of infinite groups in which any two (proper) subgroups of the same cardinality are isomorphic is described within the universe of locally graded groups. The corresponding problem for finite groups was considered by R. Armstrong (1958).