Hi Darij.
$Ind_H^G X = R[G] \otimes_{R[H]} X$ and hence there are two canonical maps:
$R[G]\otimes V \otimes W \to R[G] \otimes V\otimes R[G] \otimes W, x\otimes v\otimes w\mapsto x\otimes v\otimes 1 \otimes w$ and
$R[G]\otimes V \otimes R[G] \otimes W \to R[G] \otimes V \otimes W, x\otimes v\otimes y \times otimes w\mapsto xy \otimes v \otimes w$.
Obviously the second is a right inverse to the first. Hence the first one is injective and the second is surjective. If $R$ is a field, then there cannot be injective (surjective) maps in the other direction because the dimensions don't agree.
