Tsunekazu Nishinaka

In this note, we show that an uncountable locally free group G, and therefore every locally free group, has a free subgroup whose cardinality is the same as that of G. This result directly improves the main result in [4] and establishes the primitivity of group rings of locally free groups.

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra 317 (2007) 581-592] to the general cardinality case.

We prove that every group ring of a proper ascending HNN extension of a free group with at most countably infinite rank is primitive. (c) 2007 Elsevier Inc. All rights reserved.

We prove the following theorem: Let R be a ring, l a positive integer, and n a non-negative integer. If for each x, y is-an-element-of R, either xy = yx or xy = x(n) f(y)x(l) for some f(X) is-an-element-of X2Z[X], then R is commutative.