We prove two dual recursive decompositions as a graded $\underline{\mathrm{CH}}(M)\otimes \underline{\mathrm{CH}}(N)$-module of the Chow ring $\underline{\mathrm{CH}}(M\oplus N)$ of the direct sum of matroids. We use this to obtain a decomposition of $\underline{\mathrm{CH}}(M\oplus N)$ into irreducible $\underline{\mathrm{CH}}(M) \otimes \underline{\mathrm{CH}}(N)$-modules. The result implies a new recursive formula for the Eulerian numbers. Similarly, we find a recursive decomposition of the augmented Chow ring $\mathrm{CH}(M \oplus N)$ into $\mathrm{CH}(M)\otimes \mathrm{CH}(N)$-modules, generalizing some of the results of arXiv:2002.03341. We prove analogous decompositions of (augmented) Chow polynomials of weakly ranked posets in the sense of arXiv:2411.04070.