Cotorsion-free algebras as endomorphism algebras in $L$ - the discrete and topological cases
Göbel, Rüdiger E. ; Goldsmith, Brendan
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993), p. 1-9 / Harvested from Czech Digital Mathematics Library
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The discrete algebras $A$ over a commutative ring $R$ which can be realized as the full endomorphism algebra of a torsion-free $R$-module have been investigated by Dugas and Göbel under the additional set-theoretic axiom of constructi\-bi\-li\-ty, $V=L$. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are re\-derived in a more natural topological setting and substantial generalizations to topological algebras (which could not be handled in the previous linear algebra approach) are obtained. The results obtained are independent of the usual Zermelo-Fraenkel set theory ZFC.

Publié le : 1993-01-01
Classification:  03C60,  03E35,  16A65,  16S50,  16W80,  20K20,  20K30 
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     author = {R\"udiger E. G\"obel and Brendan Goldsmith},
     title = {Cotorsion-free algebras as endomorphism algebras in $L$ - the discrete and topological cases},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {34},
     year = {1993},
     pages = {1-9},
     zbl = {0804.16031},
     mrnumber = {1240198},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118550}
}

Göbel, Rüdiger E.; Goldsmith, Brendan. Cotorsion-free algebras as endomorphism algebras in $L$ - the discrete and topological cases. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 1-9. http://gdmltest.u-ga.fr/item/118550/

Corner A.L.S. Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 13 (1963), 687-710. (1963) | MR 0153743

Corner A.L.S. On the existence of very decomposable Abelian groups, in Abelian Group Theory, Proceedings Honolulu 1982/83, LNM 1006, Springer-Verlag, Berlin,1983. | MR 0722629

Corner A.L.S.; Göbel R. Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), 447-479. (1985) | MR 0779399

Dugas M.; Göbel R. Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc.(3) 45 (1982), 319-336. (1982) | MR 0670040

Dugas M.; Göbel R. Every cotorsion-free algebra is an endomorphism algebra, Math. Z. 181 (1982), 451-470. (1982) | MR 0682667

Dugas M.; Göbel R. Almost $\Sigma $-cyclic Abelian $p$-groups in $L$, in Abelian Groups and Modules (Udine 1984), CISM Courses and Lectures No. 287, Springer-Verlag, Wien-New York, 1984. | MR 0789809

Dugas M.; Göbel R. Torsion-free Abelian groups with prescribed finitely topologized endomorphism rings, Proc. Amer. Math. Soc. 90 (1984), 519-527. (1984) | MR 0733399

Eklof P.; Mekler A. On constructing indecomposable groups in $L$, J. Algebra 49 (1977), 96-103. (1977) | MR 0457197 | Zbl 0372.20042

Eklof P.; Mekler A. Almost Free Modules: Set-Theoretic Methods, North Holland, 1990. | MR 1055083 | Zbl 1054.20037

Fuchs L. Infinite Abelian Groups, Vol. I (1970), vol. II (1973), Academic Press, New York. | MR 0255673 | Zbl 0338.20063

Göbel R.; Goldsmith B. Essentially indecomposable modules which are almost free, Quart. J. Math. (Oxford) (2) 39 (1988), 213-222. (1988) | MR 0947502

Göbel R.; Goldsmith B. Mixed modules in $L$, Rocky Mountain J. Math. 19 (1989), 1043-58. (1989) | MR 1039542

Göbel R.; Goldsmith B. On almost-free modules over complete discrete valuation rings, Rend. Sem. Mat. Univ. Padova 86 (1991), 75-87. (1991) | MR 1154100

Jech T. Set Theory, Academic Press, New York, 1978. | MR 0506523 | Zbl 1007.03002