I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely generated left $E$-modules and similarly for $\textbf{mod}\hspace{.1 cm} Z$. We have a functor $F: \textbf{mod}\hspace{.1 cm} Z\rightarrow \textbf{mod}\hspace{.1 cm} E$ which takes $M$ to $E\otimes_Z M$, hence an induced map on (Quillen) $K_1$-groups $K_1(F): K_1(\textbf{mod}\hspace{.1 cm} Z)\rightarrow K_1(\textbf{mod}\hspace{.1 cm} E)$.

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 $K_1(\textbf{mod}\hspace{.1 cm} E)\rightarrow K_1(\textbf{mod}\hspace{.1 cm} Z)$. Information pertaining to this map would be of interest as well.