Let R be a ring and let Lγ(R) be the lattice of right ideals. We define that I ∊ Lγ(R) is a prime right ideal provided that if AB⊆I for some A, B in Lγ(R) such that AI⊆I then either A⊆I or B⊆I. Any prime ideal of a ring R is a prime right ideal and if R is commutative then an ideal is prime if and only if it is a prime right ideal. If R is a ring and a∊R, let aR={x ∊ R | x=ar for some r∊R} and aR1={x ∊ R | x = na+ar for some integer n and r ∊ R}.

J. C. Burkill [1] has defined the AP-integral whose indefinite integral is approximately continuous. An (approximately continuous) function which is approximately derivable at all points of an interval is necessarily an indefinite AP-integral of its approximate derivative.

In this note we prove the following

Theorem. If G is a nonunimodular locally compact group and 1<p<∞, then there is an open set, U, in G and there are functions, f simultaneously in every Lr(G), p≤r ≤∞, and g simultaneously in every Lq(G), 1 ≤q≤∞, such that the convolution, f * g(y), is not defined for any y in U.

It has been conjectured that if p(z) is a polynomial of degree n then the product P(z)=p(z)p'(z)p"(z)…p(n-1)(z) has at least n+1 distinct zeros unless p(z)=c(z—a)n. Professor P. Erdös who mentioned this problem in a lecture at the University of Montreal attributed it to Tiberiu Popoviciu.

The purpose of this note is to give a direct proof of Koh's main theorem in [2, p. 31] applying well-known theorems of Faith and Utumi. While doing so, we shall mention a general theorem from which Koh's theorem may be obtained as a corollary.