Let $R$ be a commutative, unital local ring with maximal ideal $m$, and residue field $k$. We can embed $k$ in its perfect closure by standard results.
My question: In general, for any such ring $R$, can we find a local ring $S$ such that:
- Its residue field is perfect, and
- There is a ring homomorphism $R\to S$ which is an injection ?
I ask because I'm dealing with local rings, and it would be handy to assume that their residue field is perfect. (Or that they can be mapped via injections to rings which have this property).
