Let A be an H-simple commutative H-module algebra, with AH = k and dimk H ≤ dimk A < ∞. We show that this implies that A # H is isomorphic to Mn(k), a central simple algebra. We apply this to characterize certain group graded algebras, algebras acted upon by a group as automorphisms, or by a nilpotent Lie algebra as derivations.

Hyadic spaces are the continuous images of a hyperspace of a compact space. We prove that every non-isolated point in a hyadic space is the endpoint of some infinite cardinal subspace. We isolate a more general order-theoretic property of hyerspaces of compact spaces which is also enjoyed by compact semilattices from which the theorem follows.

The intricacy of the problem of (2s + 1 )-factorizating Kn is determined. Some generalizations are also given.

Let R be a commutative ring with identity. An R-module M is said to be distributive if the lattice of submodules of M is distributive. We characterize such modules and study their properties.

Denote by d2 the immanant afforded by Sn and the character corresponding to the partition (2, 1n-2). If n ≥ 4, the following analog of Oppenheim's inequality is proved:

for all n-by-n positive semidefinite hermitian A and B.

If the Laurent series

is transformed to

it is shown that convergence of the former at z = 1 implies the uniform convergence of the latter on a symmetric arc of |z - 1/P| = 1/P - 1 not containing z = 1 and that the uniform convergence of the former over a symmetric arc of |z| = 1 containing z = 1 implies uniform convergence of the latter on the entire circle |z — 1/P| = 1/P — 1.

An elementary proof of the Nullstellensatz for commutative regular rings is given.

This paper constructs a class of examples to show that for torsion-free groups H with finite nilstufe v(H) = n < ∞ there can be divisible torsion groups D with v(H ⊕ D) - n + k for all k ≤ n + 1. This answers a question of Feigelstock. The construction is based on a proposition which bounds v(H ⊕ D) in terms of v(H) and rank (D).

A new concept of generic sign-stability is proposed, and a necessary and sufficient condition for this property is given. This result shows that the condition proposed by Quirk and Ruppert [12] is correct almost everywhere, and helps to clarify the counterexample presented by Jeffries [4].

A complex nilmanifold X is isomorphic to a product X ⋍ ℂp x N/┌, where N is a simply connected nilpotent complex Lie group and ┌ is a discrete subgroup of N not contained in a proper connected complex subgroup of N. The pair (N, ┌) is uniquely determined up to holomorphic group isomorphisms.

Let a be a constant, |a| = 1. We shall prove meromorphic (M) and bounded-holomorphic (BH) versions of the following prototype: (P) Let f and g be holomorphic in a domain D. Then, |f'| = |g'| in D if and only if there exist constant a, b with f = ag + b in D. (M) Let f and g be meromorphic in D. Then, |f'|/(1 + |f|2) = |g'|/(1 + |g|2) in D if and only if there exist a, b with |b| ≦ ∞ such that f = a(g - b)/(\ + g). (BH) Let f and g be holomorphic and bounded, |f| < 1, |g| < 1, in D. Then, |f'|/ (1 - |f|2) = |g'|/(1 - |g|2) in D if and only if there exist a, b with |b| < 1, such that f = a(g - b)/(1 - g).