On right pure semisimple hereditary rings and an Artin problem

Abstract

The pure semisimplicity conjecture (pss R ) 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 (pss R ) for hereditary rings R. It follows from our main results together with recent results of Herzog [16] that in order to prove (pss R ) for any R it is sufficient (and necessary) to construct an indecomposable module of infinite length over any hereditary ring R of the form ( 0 G F FMF G ), where F, G are division rings and F M G is a simple F- G-bimodule such that dim M G is finite and dim f 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.