Modules over infinite-dimensional algebras

Let A be an infinite-dimensional K-algebra, where K is a field and let B be a basis for A. We explore when K^B (the direct product indexed by B of copies of the field K) can be made into an A-module in a natural way. We call a basis B satisfying that property ‘amenable,’ and we explore when amenable bases yield isomorphic A-modules. For the latter purpose, we consider a relation, which we name congeniality, that guarantees that two different bases yield (naturally) isomorphic A-module structures on K^B. While amenability depends on the algebra structure, congeniality of bases depends only on the vector space structure and is thus independent from the specific algebra structure chosen. Among other results, we show that every algebra of countable infinite dimension has at least one amenable basis. Most of our examples will be within the familiar settings of the algebra K[x] of polynomials with coefficients in K. We show that the relation of proper congeniality (when congeniality is not symmetric) yields several natural interesting questions; among these questions we highlight those related to a natural notion of simplicity of bases. We show that the algebra of polynomials with coefficients in K has at least as many truly distinct (so-called discordant) simple bases as there are elements in the base field K.