  We give a structure theorem for inside factorial domains. As an example we
study the monoid of nonnegative integer solutions of equations of the form
$a_1x_1+\cdots +a_{r-1}x_{r-1}=a_rx_r$, with $a_1,\ldots,a_r$ positive
integers. This set is isomorphic to a simplicial full affine semigroup, and
thus it can be described in terms of its extremal rays and the Ap\'ery sets
with respect to the extremal rays.
