Primitivity testing in free group algebras via duality

Abstract Let be a field and a free group. By a classical result of Cohn and Lewin, the free group algebra 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 and an element , we give an explicit algorithm determining whether is part of some basis of . More generally, given free ‐modules , we provide algorithms determining whether is a free summand of , and whether admits a free splitting relative to . These can also be used to obtain analogous algorithms for free groups . As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free ‐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.