The concepts of Kronecker maps and Sylvester-polynomial matrix equations are proposed. Some properties of the Kroneeker maps are deduced. It is shown that many kinds of matrix equations can be formulated into the so-called Sylvester-polynomial matrix equations. With the help of the Kronecker maps, general explicit solutions of the Sylvester-polynomial matrix equations are established when the polynomial matrices possess F-primeness. For the nonsquare case, the proposed solutions can offer all the degrees of freedom represented by a free parameter matrix.