Summary: For a vector space \(V\) over the division ring \(D\), let \(\text{FEnd}_D(V)\) be the set of all \(D\)-transformations \(x\in\text{End}_D(V)\) such that \(x\) has finite rank, and let \(\text{FGL}_D(V)\) be the set of all \(g\in\text{GL}_D(V)\) such that \(g-1\) has finite rank. We study subrings \(A\) of \(\text{FEnd}_D(V)\) and the relationships of such rings to subgroups \(G\) of \(\text{FGL}_D(V)\) with the additional assumption that \(D\) is finite dimensional over its center \(K\). The division ring \(D\) is identified with \(D\cdot\text{id}_V\). With \(D\), \(A\) as above, we define \(D[A]\) to be the subring of \(\text{End}_K(V)\) generated by \(D\) and \(A\), and \(D[G]\) to be the subring of \(\text{End}_K(V)\) generated by \(D\) and \(G\). One of our ring-theoretic results is the following: if \(D[A]\) is semisimple and \(M\) is a right \(D[A]\)-module, then \(M(DA)\) is a completely reducible \(D[A]\)-module. This implies the following group-theoretic result: if \(G\leq\text{FGL}_D(V)\) and \(G\) has a local system of subgroups \(\Gamma\) such that \(H\in\Gamma\) implies that \(V\) is a completely reducible \(D[H]\)-module, then \([V,G]\) is a completely reducible \(D[G]\)-module. Several additional known results due to Meierfrankenfeld and Wehrfritz follow naturally as consequences of the ring-theoretic approach adopted here.