On the functional equation $$f\left( x \right) = \sum\limits_{j = 0}^{k - 1} {f\left( {{{\left( {x + j} \right)} \mathord{\left/ {\vphantom {{\left( {x + j} \right)} k}} \right. \kern-\nulldelimiterspace} k}} \right)} $$ over finite ringsover finite rings