A calculation of injective dimension over valuation domains
A method is presented making it possible to construct -groups with a strong theory of quasi-divisors of finite character and with some prescribed properties as subgroups of restricted Hahn groups , where are finitely atomic root systems. Some examples of these constructions are presented.
For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here "superdecomposable" means that A admits no (directly) indecomposable R-algebra summands ≠ 0 and "E(R)-algebra" refers to the property that every R-endomorphism of the R-module, A is multiplication by an element of, A.
In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if is a quasi-homeomorphism, a sober space and a continuous map, then there exists a unique continuous map such that . Let be a -space, the injection of onto its sobrification . It is shown, here, that , where is the set of all locally closed points of . Some applications are also indicated. The Jacobson prime spectrum...