For a domain R, an
R-module A is called a Baer module if ExtR1(A,T) = 0 for every torsion R-module
T. Dual to Baer modules, a torsion R-module B is called a UT-module if
ExtR1(X,B) = 0 for every torsion free R-module X. In this paper properties of these
two types of modules will be derived and characterizations of Prüfer domains,
Dedekind domains and fields will be obtained in tenns of Baer and UT-module
properties. One characterization will show the Baer modules are analogous to
projective modules in the sense that a domain R is DedekinU if and only if, over R,
submodules of Baer modules are Baer. In addition, just as a semisimple ring S
can be characterized by the property that all S-modules are injective, or,
equivalently, a11 S-modules are projective, a domain R is a field exactly if
every torsion R-module is UT or, equivalently, every torsion free R-module
is a Baer module. Further properties of these two kinds of modules will
provide sufficient conditions to bound the global dimension of a domain
R.
