Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.

The following problem due to A. Boyd, has enjoyed a certain popularity in recent months with several mathematicians. A different solution to the one given here has been given independently by R. T. Bumby and J. Spencer.

The Problem, There are n ladies, and each one of them knows an item of scandal which is not known to any of the others. They communicate by telephone, and whenever two ladies make a call, they pass on to each other, as much scandal as they know at that time. How many calls are needed before all the ladies know all the scandal?

In a recent note in the Bulletin, Murphy [5] gave a short proof that for complex m×n matrices A and B, r(A+B)=r(A)+r(B) if the rows of A are orthogonal to the rows of B and the columns of A are orthogonal to the columns of B. His proof was elegant and simple, an improvement on an earlier proof of the same result by Meyer [4].

In this note we construct a pair of regular matrices T1 and T2 such that T1 is stronger than T2, and the T2-transforms of all bounded sequences are such that the sets of limit-points are connected, while there is a bounded sequence such that the set of limit-points of its T1-transform is not connected. This provides a negative answer to a question posed by Leviatan and Lorch [3, §6, (c)].