Splitting torsion pairs over pure semisimple rings

Let R   be a left pure semisimple ring, and (D,C)(D,C) be a splitting torsion pair of R  -ind, i.e. (D,C)(D,C) is a partition of the family of all indecomposable left R  -modules such that HomR(D,C)=0HomR(D,C)=0 whenever D∈DD∈D and C∈CC∈C. Suppose further that DD contains all indecomposable injective left R-modules. We show that for each module M   in CC, the endomorphism ring of M   is a division ring and ExtR1(M,M)=0. Let W   be the direct sum of all Ext-injective modules in CC and all indecomposable projective modules in DD. If W   is endofinite, then there is an Ext-injective module in CC which is the source of a left almost split morphism in R-mod. It is also proved that W is a tilting module, and if R is hereditary, then W has a hereditary endomorphism ring. As consequences, we recover with new proofs several recent results on left pure semisimple rings R. When R is left pure semisimple hereditary indecomposable, splitting torsion pairs and tilting modules over R can be characterized using the Ext-injective partition of R-mod. In particular, when R is left pure semisimple hereditary indecomposable with only two simple modules, we give a complete description of the distribution of indecomposable left R-modules, and their Gabriel–Roiter measures.