= Simple modules over an algebra finite over its center
{title2=$\operatorname{Ann}_{Z(A)}M\text{ is maximal}$}
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 $B=R/\operatorname{Ann}_R(M)$. For $0\ne r\in B$, write generators as $m_i=r\sum_jb_{ij}m_j$. 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 $k$.
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