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Let $\varphi:R\longrightarrow S$ be a homomorphism of commutative
rings with $R$ Noetherian. We say that $\varphi$ is locally of
finite injective dimension if the injective dimension of
$S_{\frak{m}}$ as an $R_{\frak{m}}$-module is finite for every
maximal ideal $\frak{m}$ in $S$. If $R$ is a Gorenstein ring then
the identity map on $R$ is locally of finite injective dimension.
Therefore, rings which are locally of finite injective dimension
generalizes the notion of Gorenstein ring. The purpose of this
paper is to generalize some well--known results of Gorenstein
rings.
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