Ideals of injective dimension 1
Let R be an integral domain with quotient field Q and let I be a non-zero ideal of R. In this paper the author investigates conditions under which the injective dimension of I is one (i.e. Q/I is injective). This leads naturally to the conditions \(E(R/I)=Q/I\) and \(E(R/I)\subseteq Q/I\) where E(R/I) denotes the injective envelope of R/I. The results relate these conditions to properties of certain overrings of R that the author calls complemented extensions, and to properties of torsion-free liftings and coverings. A ''complemented extension'' is defined as an R-submodule A of Q which contains R for which there exists an R-submodule A' of Q with \(A+A' = Q\) and \(A\cap A' = R\). --- These are investigated in section one where it is shown that A is a flat overring of R and A' is unique. It then follows that \(Q/I\cong Q/IA\oplus Q/IA'\) so these extensions can be expected to play a role in the above questions about Q/I. It is also shown that if H is the completion of R in the R-topology, then complemented extensions of R correspond to decompositions of H into a product of two subrings. -In section two it is shown that \(inj \dim_ RI=1\) is equivalent to the following equivalent conditions on I. (1) There exists a complemented extension A of R such that \(IA\cap R=I\), \(inj \dim_ AIA=1\), and \(inj \dim_{A'}A'\leq 1.\) (2) \(R_ S\) is a complemented extension of R, \(inj \dim_{R_ S}I_ S=1\) and \(inj \dim_{R'\!_ S}R'\!_ S\leq 1\) where \(S=\{1-a| a\in I\}\) and R'\({}_ S\) is the unique complement of \(R_ S\). A similar description of when \(E(R| I)\subseteq Q/I\) holds is also given. -In section three the author shows how the properties under consideration are related to torsion-free liftings and covers of various modules. For example this section contains the result that \(inj \dim_ RI=1\) is equivalent to each of the following conditions. \((1)\quad Ext^ 1\!_ R(X,IH)=0\) for any torsion-free R-module X. (2) The exact sequence \(0\to IH\to H\otimes Q\to Q/I\to 0\) is a torsion-free lifting. (3) The exact sequence \(0\to rIH\to H\to R/rI\to 0\) is a torsion-free lifting for any non-zero \(r\in R\). Similar characterizations of \(E(R/I)=Q/I\) and \(E(R/I)\subseteq Q/I\) are also given. -In the fourth and final section the author gives applications to valuation rings, noetherian domains and h-local domains. The following is a sample of the results in this section: The following properties of R are equivalent. (1) R is an almost maximal valuation ring. (2) R is a valuation ring and \(inj \dim_ RI=1\) for every non-zero ideal I of R. (3) R has only one maximal ideal M and \(inj \dim_ RM=1.\) (4) There exists a maximal ideal M of R such that \(E(R/M)=Q/M.\) (5) There exists a maximal ideal M of R such that \(H\to R/I\) is a torsion-free cover.
scientific article; zbMATH DE number 3849380