Elementary matrix factorizations over B\'ezout domains

We study the homotopy category $\mathrm{h}\mathrm{e}\mathrm{f}(R,W)$ (and its ${\mathbb{Z}}_2$-graded version $\mathrm{H}\mathrm{E}\mathrm{F}(R,W)$) of elementary factorizations, where $R$ is a B'ezout domain which has prime elements and $W=W_0{W}_c$, where $W_0\in R^{\times }$ is a square-free element of $R$ and $W_c\in R^{\times }$ is a finite product of primes with order at least two. In this situation, we give criteria for detecting isomorphisms in $\mathrm{h}\mathrm{e}\mathrm{f}(R,W)$ and $\mathrm{H}\mathrm{E}\mathrm{F}(R,W)$ and formulas for the number of isomorphism classes of objects. We also study the full subcategory $\mathbf{h}\mathbf{e}\mathbf{f}(R,W)$ of the homotopy category $\mathrm{h}\mathrm{m}\mathrm{f}(R,W)$ of finite rank matrix factorizations of $W$ which is additively generated by elementary factorizations. We show that $\mathbf{h}\mathbf{e}\mathbf{f}(R,W)$ is Krull-Schmidt and we conjecture that it coincides with $\mathrm{h}\mathrm{m}\mathrm{f}(R,W)$. Finally, we discuss a few classes of examples.