Indecomposable injectives over the Jacobson algebra

Let [Formula: see text] be any field. In this paper, we give a complete list of the indecomposable left injective module over the Jacobson algebra [Formula: see text], i.e. the free associative [Formula: see text]-algebra on two (non-commuting) generators, modulo the single relation [Formula: see text]. This is the natural continuation of the paper of the second two authors with Gene Abrams on the characterization of the injective envelope of the simple modules over [Formula: see text].