Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ the injective hull of $l$ over $S$. Suppose that we have shown that $Hom_R(S,E_R(k))$ is an injective $S$-module. Then by the decomposition theorem of injective modules (e.g. Theorem 18.5 in Matsumura) we must have that $Hom_R(S,E_R(k)) \cong E_S(l)^r$ for some $r>0$.
Question: An identical situation arises in the proof of Theorem 3.3.7 in the last line of page 112 in Bruns and Herzog and the authors write $Hom_R(S,E_R(k))\cong E_S(k)^r$. Should it not be $E_S(l)^r$?