A prime decomposition theorem for string links in a thickened surface

We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link $\mathrm{\ell }\subset \mathrm{\Sigma }\times I$, where $\mathrm{\Sigma }$ is a compact orientable (not necessarily closed) surface other than $S^2$, can be written in the form $\mathrm{\ell }={\mathrm{\ell }}_1\mathrm{\#}\dots \mathrm{\#}{\mathrm{\ell }}_m$, where ${\mathrm{\ell }}_j,j=1,\dots ,m,$ is prime string link defined up to braid equivalence, and the decomposition is unique up to possibly permuting the order of factors in its right-hand side.