Vol. 54, No. 2, 1974

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Baer and UT-modules over domains

Ralph Peter Grimaldi

Vol. 54 (1974), No. 2, 59–72

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.

Mathematical Subject Classification 2000
Primary: 13C10
Received: 20 July 1972
Revised: 14 January 1974
Published: 1 October 1974
Ralph Peter Grimaldi