I’ll start off a little vauge: Let E be a noncommutative ring which is finitely generated over its noetherian center Z. Denote by modE the category of finitely generated left E-modules and similarly for modZ. We have a functor F:modZ→modE which takes M to E⊗ZM, hence an induced map on (Quillen) K1-groups K1(F):K1(modZ)→K1(modE).
I’m interested in situations in which this map splits. In particular, I’m interested when E is a nice endomorphism ring and Z is the normalization of a Cohen-Macaulay local ring of finite type (this certainly occurs: See Proposition 2.15 of http://arxiv.org/pdf/1401.3000.pdf). Results in positive dimension appreciated!
Edit: Since E is finitely generated over Z, we also have a map induced by restriction of scalars K1(modE)→K1(modZ). Information pertaining to this map would be of interest as well.