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A solution to the Baer splitting problem. (English) Zbl 1136.13005

Let \(R\) be a commutative domain. A module \(B\) is called a Baer module if \(\mathrm{Ext}^1_R(B,T)=0\) for every torsion \(R\)-module \(T\). It was proved by R. Baer [Ann. Math. (2) 37, 766–781 (1936; Zbl 0015.20202)] that every countable generated Baer abelian group must be free. I. Kaplansky studied these modules over commutative domains using homological algebra techniques and showed that are flat modules of projective dimension at most one [Arch. Math 13, 341–343 (1962; Zbl 0108.26302)]. He also asked if Baer modules are free. This question was affirmatively answered by P. Griffith for abelian groups [Trans. Am. Math. Soc. 139, 261–269 (1969; Zbl 0194.05301)]. Grimaldi [Ph. Thesis, New Mexico State University, (1972)] proved that Baer modules over Dedekind domain are projective. P. Eklof and L. Fuchs [Ann. Mat. Pura Appl. 150, 363–373 (1988; Zbl 0654.13014)], using set theoretic methods, established that Baer modules over valuation domains are free. More recently, P. Griffith [Ill. J. Math. 47, 237–250 (2003; Zbl 1033.20062)] solved the problem for local noetherian regular domain. The authors show that every countable generated Baer module over a commutative domain is projective. The idea is to translate the Baer and projective condition into a Mittag-Leffler condition on inverse limit, then studing the closure properties of the class of modules satisfying this condition they are able to see that the ring belongs to this class that turn out to be equivalent to the projectivity condition.

MSC:

13C10 Projective and free modules and ideals in commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
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References:

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