Baer Modules Research Articles

On Whitehead modules

It is proved that it is consistent with ZFC + GCH that, for any reasonable ring R, for every R-module K there is a non-projective module M such that Ext R 1( M, K) = 0; in particular, there are Whitehead R-modules which are not projective. This is generalized to show that it is consistent that, for certain rings R, there are Whitehead R-modules which are not the union of a continuous chain of submodules so that all quotients are small Whitehead R-modules. An application to Baer modules is also given: it is proved undecidable in ZFC + GCH whether there is a single test module for being a Baer module.

Baer Modules over Domains

For a commutative domain $R$ with $1$, an $R$-module $B$ is called a Baer module if $\operatorname {Ext} _R^1(B,T) = 0$ for all torsion $R$-modules $T$. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over $h$-local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.

Baer modules over domains

For a commutative domain R R with 1 1 , an R R -module B B is called a Baer module if Ext R 1 ⁡ ( B , T ) = 0 \operatorname {Ext} _R^1(B,T) = 0 for all torsion R R -modules T T . The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over h h -local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.

Baer modules over valuation domains

A module B over a commutative domain R is said to be a Baer module if Ext R 1 (B, T)=0for all torsion R-modules T. The case in which R is an arbitrary valuation domain is investigated, and it is shown that in this case Baer modules are necessarily free. The method employed is totally different from Griffith's method for R=Z which breaks down for non-hereditary rings.

Baer modules

In an abelian category with a torsion theory an object B is called a Baer object if Ext( B, T) = 0 for all torsion objects T. The Baer C [ξ]-modules are precisely the free C [ξ]-modules (P. Griffith). The modules over finite-dimensonal hereditary algebras of tame type have some properties analogous to C [ξ]-modules. However, in this paper it is shown that Baer modules over such algebras are not necessarily free or even pure-projective. A characterisation of Baer modules of countable rank over a particular Artin algebra is also given: A system of countable rank is Baer if and only if ( V, W) = ∪ n = 1 ∞ ( V n , W n ), where ( V n , W n ) is a finite-dimensional subsystem of ( V n + 1 , W n +1 ) and (V n +1, W n +1) (V n, W n) is torsion-free for n = 1, 2,…. Baer systems of arbitrary rank are also discussed.

Baer and UT-modules over domains

For a domain R, an it-module A is called a Baer module if Ext £ 04, T) = 0 for every torsion /^-module T. Dual to Baer modules, a torsion it-module B is called a UT-moάule if Ext \(X, B) = 0 for every torsion free it-module X. In this paper properties of these two types of modules will be derived and characterizations of Priίfer domains, Dedekind domains and fields will be obtained in terms of Baer and l/T-module properties. One characterization will show the Baer modules are analogous to projective modules in the sense that a domain R is Dedekind 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 5-modules are injective, or, equivalently, aU 5-modules are projective, a domain R is a field exactly if every torsion it-module is UT or, equivalently, every torsion free it-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. 0. Historical Note. The concept of a Baer module goes back to 1936 when R. Baer, in [1], proposed the problem asking for a complete characterization of all abelian groups G such that Extz (G, T) = 0, for all torsion abelian groups T. At that time he showed that any such abelian group must be torsion free, and free if it had countable rank. Then in 1959 R. Nunke, in [10], extended these results to modules over a Dedekind domain, proving that such a module was again torsion free, and projective if it had countable rank. Finally in 1969 P. Griffith, in [5], completely solved Baer's problem for abelian groups, and showed that any such abelian group, now called a Baer group or 2?-grouρ, must be free. In [6], the author extended Griffith's techniques to modules over a Dedekind domain, showing that if A is a Baer module over a Dedekind domain then A is projective. The major adjustments needed in this transition from abelian groups to modules over a Dedekind domain were accomplished by means of Corollary 2 on p. 279 of [12], Theorems 3 and 5 in [8], Lemma 8.3 and Theorem 8.4 in [10], as well as the exposition on ideals and valuations in § 18, 19 of 1. Preliminaries. Unless additional restrictions are stated, in this paper R denotes an arbitrary integral domain: that is, a commutative ring with 1 having no zero divisors. The quotient field oίR will be denoted by Q.