Wilbur Whitten has pointed out to me that there exist invertible links amongst my family of allegedly non-invertible links [3], and a letter from Neville Smythe has confirmed that the error lies in the third paragraph from the bottom of page 106. Professor Smythe also suggested that the use of F-isotopy was possibly a red herring and, accordingly, I have devised a simpler proof of the existence of infinitely many non-invertible prime links in S3 (although Wilbur Whitten's paper [4] makes such a proof redundant).

Let X be a commutative locally convex Hausdorff topological algebra with identity over a non-trivially valued field F. Let Mc denote the continuous nontrivial homomorphisms of X into F and M the set of all maximal ideals of X. If the spectrum of each element x in X is the set of scalars {f(x) | f ∈ Mc}, it is shown that the singular elements of X are weakly dense in X if and only if M is an infinite set.

Let denote the variety of all abelian groups and, for each prime p, let p be the variety of all elementary abelian p-groups. Let be a subvariety of a product of (finitely many) varieties each of which is either soluble or Cross.

It is well-known that every finite ring with non-zero-divisors has order not exceeding the square of the order n of its left zero-divisor set. Unital rings whose order is precisely n2 have been described already. Here we discuss finite rings with relatively larger zero-divisor sets, namely those of order greater than n3/2. This is achieved by describing the class of all finite rings with left composition length two at most, and using a theorem relating the left composition length of a finite ring to the size of its left zero-divisor set.

The sub-near-field structure of finite near-fields is analogous to the sub-field structure of finite fields. A finite near-field of order pln contains a unique sub-near-field of order pλ if and only if λ divides ln.

Let N be a subgroup of the torsion-free abelian group G. Then a partial order for N is contained in one, two or uncountably many full orders for G, and a full order for nonzero N is contained in one or uncountably many full orders for G.

It is proved that, for any m given distinct real numbers a1, …, am, there exist transcendental entire functions f(z) at most of order m for which all the values

are rational integers.

Let p denote a prime, G a finite p′-group and A an elementary atelian group of operators on G. Suppose that A has order p3 and that if w ∈ A# then CG(w) is nilpotent. It is proved that G is nilpotent.