The New Results in Injective Modules

In this paper, we introduce and clarify a new presentation between the divisible module and the injective module. Also, we obtain some new results about them.

Equivalence elementaire et decidabilite pour des structures du type groupe agissant sur un groupe abelien

We prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.Two applications come immediately from this result:First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.

Simple Divisible Modules Over Integral Domains

An R-module is a simple divisible module if it is a nonzero divisible module that has no proper non-zero divisible submodules. We study simple divisible modules and their endomorphism rings, give some examples and determine all simple divisible modules over some classes of rings.

On Divisibility and Injectivity

In the category of abelian groups the concepts of divisible group and injective group coincide. In (4) this is generalized to modules over an integral domain and it is proved for a (commutative) integral domain that the concepts of divisible module and injective module coincide if and only if the ring is hereditary if and only if the ring is a Dedekind domain.In (8) the assumption of commutativity is dropped and the ring is assumed to have a one-sided field of quotients. The result obtained is essentially the same as in the commutative case; see the theorem following 6.2. In (13) the requirement that the ring have no zero-divisors is also dropped and the ring is assumed to possess what we have called an Ore ring (see the definition following 6.2). The result obtained is the equivalent of parts (a) and (b) of our 6.13.