  We study commutative algebra arising from generalised Frobenius numbers. The
$k$-th (generalised) Frobenius number of natural numbers $(a_1,\dots,a_n)$ is
the largest natural number that cannot be written as a non-negative integral
combination of $(a_1,\dots,a_n)$ in $k$ distinct ways. Suppose that $L$ is the
lattice of integers points of $(a_1,\dots,a_n)^{\perp}$. Taking cue from the
concept of lattice modules due to Bayer and Sturmfels, we define generalised
lattice modules $M_L^{(k)}$ whose Castelnuovo-Mumford regularity captures the
$k$-th Frobenius number of $(a_1,\dots,a_n)$. We study the sequence
$\{M_L^{(k)}\}_{k=1}^{\infty}$ of generalised lattice modules providing an
explicit characterisation of their minimal generators. We show that there are
only finitely many isomorphism classes of generalized lattice modules. As a
consequence of our commutative algebraic approach, we show that the sequence of
generalised Frobenius numbers forms a generalised arithmetic progression. We
also construct an algorithm to compute the $k$-th Frobenius number.
