We show here that every $m$-dimensional semiring module $M$ over an idempotent semiring $S$ with strongly independent basis can be embedded into $S^m$, and provide an algebraic invariant - the $\Lambda$-matrix - which characterises the isomorphy class of $M$. The strong independence condition also yields a significant improvement to the Whitney embedding for tropical torsion modules published earlier [LAA 435, 1786-1795, 2011]. We also show that the strong independence of the basis of $M$ is equivalent to the unique representation of elements of $M$. Numerous examples illustrate our results, and a fast test for strong independence of the columns of a matrix is provided.