 Simple modules over an algebra finite over its center

If $A$ is a finite module over a commutative central subring $R$ and $M$ is a simple left $A$-module, then $M=Am$ is finite over $R$. A central element not annihilating $M$ acts bijectively, since its kernel and image are $A$-submodules. Put $B=R/{\mathrm{Ann}}_R(M)$. For $0\ne r\in B$, write generators as $m_i=r{\sum }_j{b}_{ij}{m}_j$. The adjugate matrix identity shows $\det (I-r(b_{ij}))$ annihilates $M$, hence is zero in $B$. Since this determinant has form $1-rq$, $r$ is invertible. Thus $B$ is a field. If $R$ is a finite-type algebra over an algebraically closed field $k$, the residue field is $k$, and every such simple module is finite-dimensional over $k$.