Each semiprime ring admits a unique projectable, strongly projectable, laterally complete and orthocomplete hull. Almost all of the theory for X–hulls of lattice-ordered groups in Paul Conrad, “The hulls of representable l-groups and f-rings”, J. Austral. Math. Soc. 16 (1973), 385–415, has a counterpart for semiprime rings. The proofs of these results will appear elsewhere. They come in a large part directly from the corresponding theory for lattice-ordered groups. There is also a feedback from the rings to the groups.

In the study of the theory of rings, matrix rings, group rings, algebras, and so on, play a very important role. However, the analogous systems may not exist in the theory of near-rings. Recently Ligh obtained a necessary and sufficient condition for the set of n × n matrices with entries from a near-ring to be a near-ring. This opens the door for the study of other structures such as group near-rings, algebras, and so on. In this paper we initiate a study of the basic properties of pseudo-distributive near-rings, which is exactly the class of near-rings needed to carry out the construction of matrix near-rings, group near-rings, polynomials with near-ring coefficients, and so on.

A constructive proof of a theorem of Worrell and Wicke, which states that every open covering of a θ-refinable space has a σ-distributively point-finite open refinement that covers the space minimally, is presented. Spaces for which every open covering has an open refinement which covers the space minimally are characterized by the use of discrete closed collections and some open questions relating to spaces of this type are included.

In 1965, D.L. Berman established an interesting divergence theorem concerning Hermite-Fejér interpolation on the extended Chebyshev nodes. In this paper it is shown that this phenomenon is not an isolated incident. A similar divergence theorem is proved for a higher order interpolation process. The paper closes with a list of several related open problems.

Let S be a compact semigroup and P(S) the set of probability measures on S. Suppose P(S) has zero θ and define a measure μ ε P(S) nilpotent if μn → θ. It is shown that any measure with support containing that of θ is nilpotent, and the set of nilpotent measures is convex and dense in P(S). A measure μ is called mean-nilpotent if (μ + μ2 + … + μn)/n → θ, and can be characterized in terms of its support.

The object of this paper is to prove two new results on the nature of the spectrum of a class of nonlinear elliptic eigenvalue problems. In the first case, sufficient conditions are given for which the spectrum is bounded and, in the second case, conditions are given for which the spectrum is open.

There is a small error in the proof of Lemma 1 [1], namely, that there may not be a subsequence of the sequence {Φn} converging to Φ in the weak*-topology. Thus equation (5) should read: In view of continuity of Pk we have

The same error occurs in the proof of Theorem 1 and Lemma 2 also. Therefore, equation (7) should read

and this leads to lim . Similarly, equation (8) should read

which implies that Ψn (x) → ‖x‖. Then there is no need of taking a subsequence.

All the same the results are not affected by the above error.

In [1] I made a Remark which, when subsequently challenged by Anne Koehier, I was unable to justify. While writing up my PhD thesis, I tried to both prove and disprove the statement but failed on both counts. However, I did reduce it to a highly plausible statement which is presented here.

Professor Edward T. Ordman has pointed out to the author that the proof of Theorem 2.2 in [2] is incorrect. The theorem is, in fact, correct and was proved by Graev [1] The incorrect proof presented in my paper has been modified by Ordman [3] to provide a much simpler proof than Graev's in a special case.