Primitivity Testing in Free Group Algebras via Duality

Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\le K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces.