GENERALIZED DEDEKIND DOMAINS AND THEIR INJECTIVE-MODULES

We prove that for a commutative integral domain R the following conditions are equivalent: (a) R is a Prüfer domain with no non-zero idempotent prime ideals; (b) there is a one to one correspondence between prime ideals in R and isomorphism classes of indecomposable injective R-modules, and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. This result allows us to study and describe injective modules over generalized Dedekind domains. Furthermore, we show that a partially ordered set is order isomorphic to the spectrum of a generalized Dedekind domain if and only if it is a Noetherian tree with a least element. © 1994.