Gluing Associated Prime Ideals of Small Height

Let $B$ be a local (Noetherian) ring and suppose that $B$ has $n$ associated prime ideals where $n\ge 2$. We identify sufficient conditions for there to exist a local (Noetherian) subring $S$ of $B$ such that $S$ and $B$ have the same completion and $S$ has exactly $n-1$ associated prime ideals. We include applications and consequences of this result.