  We study the homotopy category $\mathrm{hmf}(R,W)$ of matrix factorizations
of non-zero elements $W\in R^\times$, where $R$ is an elementary divisor
domain. When $R$ has prime elements and $W$ factors into a square-free element
$W_0$ and a finite product of primes of multiplicity greater than one and which
do not divide $W_0$, we show that $\mathrm{hmf}(R,W)$ is triangle-equivalent
with an orthogonal sum of the triangulated categories of singularities
$\mathrm{D}_{\mathrm sing}(A_n(p))$ of the local Artinian rings
$A_n(p)=R/\langle p^n\rangle$, where $p$ runs over the prime divisors of $W$ of
order $n\geq 2$. This result holds even when $R$ is not Noetherian. The
triangulated categories $\mathrm{D}_{\mathrm sing}(A_n(p))$ are Krull-Schmidt
and we describe them explicitly. We also study the cocycle category
$\mathrm{zmf}(R,W)$, showing that it is additively generated by elementary
matrix factorizations. Finally, we discuss a few classes of examples.
