For an algebra, \(R\), the authors investigate the property, which they call FCR, saying that any finite-dimensional representation is completely reducible and that each element of the algebra acts non-zero on some finite-dimensional representation. As an example of such algebra one can take the universal enveloping algebra of a semisimple finite-dimensional complex Lie algebra. Let \(R=A\oplus V\), where \(A\) is a subalgebra such that \(R\) is finitely generated as an \(A\)-left module and \(V\) is an \(A\)-bimodule. The authors show that if \(R\) satisfies FCR, then so does \(A\).
This result is an easy generalization of an elementary result, also presented in the paper, stating that a subring \(A\) of a semisimple ring is semisimple if it admits a complement, which is left and right stable under \(A\). In fact, if \(\text{char }R\neq 2\), it suffices to assume that the complement is stable under commutation with \(A\).