On right pure semisimple hereditary rings and an Artin problem Original Research Article

The pure semisimplicity conjecture (pssR) stated below is studied in the paper mainly for hereditary rings R. One of our main results is Theorem 3.6 containing various conditions which are equivalent to the conjecture (pssR) for hereditary rings R. It follows from our main results together with recent results of Herzog [16] that in order to prove (pssR) for any R it is sufficient (and necessary) to construct an indecomposable module of infinite length over any hereditary ring R of the form (image), where F, G are division rings and FMG is a simple F-G-bimodule such that dim MG is finite and dimf M is infinite (see Corollary 5.1). Moreover, the existence of a counterexample R to the pure semisimplicity conjecture is equivalent to a generalized Artin problem for division rings (see 4.3–4.6), which is much more difficult than the Artin problem for division ring extensions solved by Cohn in [5] and by Schofield in [20]. It may frighten people of finding an easy solution to the pure semisimplicity problem. On the other hand, it is concluded in Section 5 that studying generalized Artin problems can help solve the pure semisimplicity conjecture.