The classical Seifert-van Kampen theorem expresses the fundamental group of a union of topological spaces as an amalgamated free product. An amalgamated free product structure on the fundamental group of a manifold can be realized by a codimension 1 submanifold, by geometric transversality. The talk will describe an algebraic analogue of transversality which decomposes modules and quadratic forms over an amalgamated free product of rings.