  Using Popa's deformation/rigidity theory, we investigate prime decompositions
of von Neumann algebras of the form $L(\mathcal{R})$ for countable probability
measure preserving equivalence relations $\mathcal{R}$. We show that
$L(\mathcal{R})$ is prime whenever $\mathcal{R}$ is nonamenable, ergodic, and
admits an unbounded 1-cocycle into a mixing orthogonal representation weakly
contained in the regular representation. This is accomplished by constructing
the $\textit{Gaussian extension }\tilde{\mathcal{R}}$ of $\mathcal{R}$ and
subsequently an s-malleable deformation of the inclusion $L(\mathcal{R})
\subset L(\tilde{\mathcal{R}})$. We go on to note a general obstruction to
unique prime factorization, and avoiding it, we prove a unique prime
factorization result for products of the form $L(\mathcal{R}_1) \otimes
L(\mathcal{R}_2) \otimes \cdots \otimes L(\mathcal{R}_k)$. As a corollary, we
get a unique factorization result in the equivalence relation setting for
products of the form $\mathcal{R}_1 \times \mathcal{R}_2 \times \cdots \times
\mathcal{R}_k$. We finish with an application to the measure equivalence of
groups.
