a central theme in the theory of von neumann algebras is to determine all possible tensor product decompositions of a given factor. i will present a recent joint work with yusuke isono where we use the rigidity of full factors and a new flip automorphism approach in order to study this problem. among other things, we show that a separable full factor admits at most countably many tensor product decompositions (up to stable unitary conjugacy). we also establish new primeness and unique prime factorization results for crossed products coming from compact actions of irreducible higher rank lattices (e.g. $sl_n(\mathbb{z})$ for $n>2$) as well as noncommutative bernoulli shifts with arbitrary base (not necessarily amenable).