Continuous homomorphisms and rings of injective dimension one - Chang, Shou-Te; Huang, I-Chiau (MATHEMATICA SCANDINAVICA, 2012)
Let S be an R-algebra and a be an ideal of S. We define the continuous hom functor from R-mod to S-mod with respect to the 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 a. Furthermore, if S is 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.