Continuous homomorphisms and rings of injective dimension one

Let $S$ be an $R$-algebra and $\mathfrak a$ be an ideal of $S$. We define the continuous hom functor from $R$-mod to $S$-mod with respect to the $\mathfrak a$-adic topology on $S$. We show that the continuous hom functor preserves injective modules iff the ideal-adic property and ideal-continuity property are satisfied for $S$ and $\mathfrak a$. Furthermore, if $S$ is $\mathfrak a$-finite over $R$, we show that the continuous hom functor also preserves essential extensions. Hence, the continuous hom functor can be used to construct injective modules and injective hulls over $S$ using what we know about $R$. Using the continuous hom functor we can characterize rings of injective dimension one using symmetry for a special class of formal power series subrings. In the Noetherian case, this enables us to construct one-dimensional local Gorenstein domains. In the non-Noetherian case, we can apply the continuous hom functor to a generalized form of the $D+M$ construction. We may construct a class of domains of injective dimension one and a series of almost maximal valuation rings of any complete DVR