Serre [5] has recently proved a general theorem about projective modules over commutative rings. This theorem has the following consequence : If 7T is a finite abelian group, any finitely generated projective module over the integral group ring Zir is the direct sum of a free module and an ideal of Zir. The question naturally arises as to whether this result holds for nonabelian groups x. Serre...

We consider structures which are FA-presentable. It is known that an FA-presentable finitely generated group is virtually abelian; we strengthen this result by showing that an arbitrary FA-presentable group is locally virtually abelian. As a consequence, we prove that any FA-presentable ring is locally finite; this is a significant restriction and allows us to say a great deal about the structu...

Let Γ be a limit group, S ⊂ Γ a non-trivial subgroup, and N the normaliser of S. If H1(S,Q) has finite Q-dimension, then S is finitely generated and either N/S is finite or N is abelian. This result has applications to the study of subdirect products of limit groups. Limit groups (or ω-residually free groups) have received a good deal of attention in recent years, largely due to the work of Z. ...

An order is a commutative ring of which the additive group is a finitely generated free abelian group, and a graded order is an order that is provided with a grading by some abelian group. Examples are provided by group rings of finite abelian groups over rings of integers in number fields. We generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the ...

Let $G$ be an abelian group and $S$ a given multiplicatively closed subset of commutative $G$-graded ring $A$ consisting homogeneous elements. In this paper, we introduce study $S$-Noetherian modules which are generalization modules. We characterize in terms For instance, $A$-module $M$ is if only $S$-Noetherian, provided finitely generated countable. Also, generalize some results on Noetherian...

where {ΓkG}k≥1 is the lower central series (LCS) of the group, defined, inductively, by Γ1G = G and Γk+1G = (ΓkG,G), and the Lie bracket [x, y] is induced from the group commutator (x, y) = xyxy. Many properties of a group are reflected in properties of its associated graded Lie algebra. For instance, if G is finitely generated, then the abelian groups grk(G) are also finitely generated; their ...