We reduce the problem of impulse elimination via state feedback in singular differential equations to algebra. Our results are developed for systems over an arbitrary Hermite domain. We show that the established theories for the time-invariant and the real analytic time-varying settings can be unified in this way. Besides the constant and real analytic functions, several other function rings are considered. Our algebraic theory is applied to these cases, providing solutions to the impulse elimination problem for classes of systems not previously studied. In particular, our work allows the restriction of the feedback matrix to certain function rings.