Kulumani M. Rangaswamy

Infinite rank Butler groups

A torsion-free abelian group G is said to be a Butler group if Bext(C, T) = 0 for all torsion groups T. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group G satisfies the T.E.P. over a pure subgroup H if and only if H is decent in G in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called B2-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ^-group. We show under ( V = L) that this is indeed the case for Butler groups of rank Nt. On the other hand it is shown that, under ZFC, it is undecidable whether a group B for which Bext( B, T) = 0 for all countable torsion groups T is indeed a B2-group.

Butler groups of infinite rank. II

A torsion-free abelian group G is called a Butler group if Bext(G, T) = 0 for any torsion group T. We show that every Butler group G of cardinality ltl is a B2-group; i.e., G is a union of a smooth ascending chain of pure subgroups G0 where G+ = Ga + B0, B,0 a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding N., is a B2-group. Moreover, we are able to prove that the derived functor Bext2 (A, T) = 0 for any torsion group T and any torsion-free A with JAI < ,,. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality < Nt, are B2-groups.

On prime nonprimitive von Neumann regular algebras

Let E be any directed graph, and K any field. We classify those graphs E for which the Leavitt path algebra LK(E) is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.

On Generators of Two-Sided Ideals of Leavitt Path Algebras over Arbitrary Graphs

Let E be an arbitrary graph and K be any field. For any nongraded ideal I of the Leavitt path algebra L K (E), we give an explicit description of a set of generators of I. This leads to a number of consequences, most importantly, every two-sided ideal of L K (E) can be equipped with a set of mutually orthogonal generators. An interesting consequence is that every finitely generated ideal of L K (E) must be a principal ideal. In particular, if E is a finite graph, then every ideal of L K (E) must be principal. As an application, we compute the cardinality of the set of all nongraded ideals of L K (E) for an arbitrary graph E. From this we obtain conditions under which L K (E) has at most finitely many distinct ideals.

TWO-SIDED CHAIN CONDITIONS IN LEAVITT PATH ALGEBRAS OVER ARBITRARY GRAPHS

Let E be any directed graph, and K be any field. For any ideal I of the Leavitt path algebra LK(E) we provide an explicit description of a set of generators for I. This description allows us to classify the two-sided noetherian Leavitt path algebras over arbitrary graphs. This extends similar results previously known only in the row-finite case. We provide a number of additional consequences of this description, including an identification of those Leavitt path algebras for which all two-sided ideals are graded. Finally, we classify the two-sided artinian Leavitt path algebras over arbitrary graphs.

On torsion-free abelian

ABSTRACT. It is shown that a knice subgroup with cardinality Ni, of a torsion-free completely decomposable abelian group, is again completely de- composable. Any torsion-free abelian fc-group of cardinality H„ has balanced projective dimension s}. G(s*) is the subgroup generated by the set{x G G(s): J2pepi\x\p ~ sp) s unbounded}. Two height sequences (sp) and (tp)are said to be equivalent if YlPep \sp ~ ?p\ 18 finite.

k

Abstract If is the union of a smooth strictly ascending chain of B 2-subgroups G α, then, in the case when κ < ℵω, a criterion is established under which Gbecomes a B 2-group. This criterion is dependent on a new class of torsion-free groups and generalizes earlier criteria for freeness established by Paul Hill. The result of S. Shelah and others establishing that, for a weakly compact cardinal κ “κ-free” implies “free”, is extended to the case of B 2-groups.

-groups

Let

Smooth unions of Butler groups

E

On intersections of two-sided ideals of Leavitt path algebras

be an arbitrary directed graph and let