In this note we give an example of an orthocomplemented modular lattice whose completion by cuts is not orthomodular. This solves negatively Problem 36, p. 131, in G. Birkhoff: Lattice theory (3rd edition).

A theorem of Gaschütz is used to prove: Let τ be a homomorphism of the distributively generated near-ring R with identity element and descending chain condition for left modules, τ have finite kernel, and U(R) be the group of units of R; then U(Rτ) = U(R)τ.

Furthermore it is shown that the finiteness condition for ker τ can be dropped in the case of R being a ring.

A basic problem in the theory of Lie algebra extensions concerns a given homomorphism X of a Lie algebra L into the Lie algebra of outer derivations of a Lie algebra B. In analogy with the theory of group extensions, Mori and HochschiId developed the concept of an obstruction to X being the homomorphism defined by some Lie algebra extension of B by L. This note considers an alternative approach to this theory, which is particularly simple when applied to the problem of realizing arbitrary three-cohomology classes of L as obstructions. The approach is analogous to one for groups, which was given recently by Gruenberg.

A ring R is π-regular (periodic) if for each element x of R there is n = n(x) SO that xn = xn.a.xn (xn = xn.1.xn) (a depending on x). Let R be an algebraic algebra over a commutative ring F With identity. In this paper we prove that if every π-regular image of the ring F is periodic, then R is periodic. This result applies in particular to the algebraic rings R (over the integers) considered by Drazin and to the algebraic algebras R over algebraically prime fields. It extends a result of Drazin on torsin-free algebraic rings and a generalization by this author of Drazin's result.

In the present note the equation yn = x1-m ym is reduced, under appropriate conditions, to a quadratic autonomous system of differential equations in the plane. In pursuance of this new approach, the main geometric features of this autonomous system are determined and a method of solving it is outlined.

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

We show that, whenever m, n are coprime, each subvariety of the abelian-by-nilpotent variety has a finite basis for its laws. We further Show that the just non-Cross subvarieties of are precisely those already known.

A variety of groups has the amalgam embedding property if every amalgam of two -groups can be embedded in a -group. In this note the author proves that if is a variety of exponent O which satisfies a law W(x1, X2,…, xt) but not W(x1, x2,… xt) then does not have the amalgam embedding property.

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

Certain theorems which are already known show that if a partially balanced incomplete block design with suitable parameters exists then there is a (V, K, Λ)-graph. We prove that the existence of such a graph is in fact equivalent to the existence of a certain partially balanced design. The known necessary conditions for (V, K, Λ)-graphs then follow from well-known necessary conditions for designs.