The isomorphism problem for basic modules and the divisibility profile of the algebra of polynomials

While mutual congeniality of bases is known to guarantee that basic modules from so-related bases are isomorphic, the question of what can be said about isomorphism of basic modules in general has remained open. We show that, for some algebras, basic modules may be non-isomorphic. We also show that it is possible, for some algebras, for all basic modules to be isomorphic, regardless of congeniality. In the process, and as a byproduct, we introduce the notion of domains of divisibility of modules over arbitrary rings. The mechanism employed here to differentiate non-isomorphic basic modules is by showing that they have different domains of divisibility. Domains of divisibility measure how divisible a module can be; for those cases when divisibility is equivalent to injectivity, domains of divisibility provide a way to gauge injectivity of modules as an alternative to the domains of injectivity and other mechanisms in the literature. We focus on the algebra of polynomials with one variable, and observe that F[x] has a full divisibility profile for any field F. When F is algebraically closed, we see that the direct product and the direct sum of all Pascal basic modules have the smallest divisibility domain. We also analyze the diversity of the family of basic modules as we explore how many of those divisibility domains correspond to basic modules. We show that, for an arbitrary infinite field F, F[x] has infinitely many pairwise non-isomorphic basic modules, the divisibility profile of F[x] is complete, and for an algebraically closed field F, the collection of basic modules has any subset S of F with a non-empty at most countable complement, the set \(\{ x + \alpha | \alpha \notin S \}\) as a domain of divisibility for some basic F[x]-module.