Topological characterizations of amenability and congeniality of bases

<div>We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.</div><div><p>A basis B over an innite dimensional F-algebra A is called amenable if F^B, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.</p><p>(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.</p></div>

Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial

If C = C(R) denotes the center of a ring R and g(x) is a polynomial in C[x], Camillo and Simón called a ring g(x)-clean if every element is the sum of a unit and a root of g(x). If V is a vector space of countable dimension over a division ring D, they showed that end DV is g(x)-clean provided that g(x) has two roots in C(D). If g(x) = x – x2 this shows that end DV is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that end RM is g(x)-clean for any semisimple module M over an arbitrary ring R provided that g(x) ∈ (x – a)(x – b)C[x] where a, b ∈ C and both b and b – a are units in R.

Independence and consistency proofs in quadratic form theory

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.

On the stability of barrelled topologies, III

Let E be a barrelled space with dual F ≠ E*. It is shown that F has uncountable codimension in E*. If M is a vector subspace of E* of countable dimension with M ∩ F = {o}, the topology τ(E, F+M) is called a countable enlargement of τ(E, F). The results of the two previous papers are extended: it is proved that a non-barrelled countable enlargement always exists, and sufficient conditions for the existence of a barrelled countable enlargement are established, to include cases where the bounded sets may all be finite dimensional. An example of this case is given, derived from Amemiya and Kōmura; some specific and general classes of spaces containing a dense barrelled vector subspace of codimension greater than or equal to c are discussed.