# Wehrfritz B. A. F.

### Finitary groups and Krull dimension over the integers

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1310–1325

Let $M$ be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup $$F_\infty Aut M = \{ g \in Aut M: M(g - 1) is\;a \;minimax\; group\}$$ of the automorphism group $Aut M$ of $M$. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the common behavior of the finitary and the Artinian-finitary subgroups of $Aut M$, are certainly stronger than one might reasonably expect. Our main focus is on residual properties and unipotence.

### Finitary and Artinian-Finitary Groups over the Integers $ℤ$

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 753-763

In a series of papers, we have considered finitary (that is, Noetherian-finitary) and Artinian-finitary groups of automorphisms of arbitrary modules over arbitrary rings. The structural conclusions for these two classes of groups are really very similar, especially over commutative rings. The question arises of the extent to which each class is a subclass of the other.

Here we resolve this question by concentrating just on the ground ring of the integers ℤ. We show that even over ℤ neither of these two classes of groups is contained in the other. On the other hand, we show how each group in either class can be built out of groups in the other class. This latter fact helps to explain the structural similarity of the groups in the two classes.

### Cohn's embedding of an enveloping algebra into a division ring

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 729–735

Let F be a field, L a Lie F-algebra and U=U(L) the universal enveloping algebra of L. In [1] Cohn constructs an embedding of U into a division ring. Recently there has been interest in this specific division ring in connection with matrix groups and matrix rings [2–4]. Cohn's construction is less than direct and it seemed useful to have a very explicit description of D, at least for the benefit of group theorists.

### Some soluble groups of finite rank and some related matrix groups

Ukr. Mat. Zh. - 1991. - 43, № 7-8. - pp. 894–901