commutative algebra – special case of isomorphism $ S otimes_R mathrm {Hom} _R (M, N) simeq mathrm {Hom} _S (S otimes_R M, S otimes_R N) $

So we have the S-module isomorphism $$ S otimes_R mathrm {Hom} _R (M, N) simeq mathrm {Hom} _S (S otimes_R M, S otimes_R N) $$ given that $ S $ is flat over $ R $ and $ M $ will finally be presented. Even though $ S $ is not flat or $ M $ is finally presented, there is at least $ S $Homomorphism between the modules.

What I am considering is a limitation of isomorphism. To let $ phi in mathrm {End} _R (M) $, $ 1 otimes_R phi $ will designate endomorphism of $ S otimes_R M $, $$ S otimes_R R[phi] Simeq S[1otimes_R phi]$$
Is the above isomorphism without the condition that $ S $ is flat over $ R $ or that $ M $ is finally presented?

In the face of that $ S $ is flat over $ R $ and $ M $ is finally presented, $ S otimes_R R[phi]$ is a module of $ S otimes_R mathrm {End} _R (M) $ and the isomorphism in question follows easily.

It is obvious that the induced homomorphism is surjective, but I can not verify exactly if it is injective.

  1. Does Isomophism have no flatness or finite presence?
  2. If the isomorphism holds, what is this structural difference between $ R[phi]$ and the whole $ mathrm {End} _R (M) $ that makes isomorphism possible on one side and not on the other?

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