Previous |  Up |  Next


cotorsion-free; endomorphism algebra; axiom of constructibility; Zermelo-Fraenkel set theory
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.
[1] Corner A.L.S.: Every countable reduced torsion-free ring is an endomorphism ring. Proc. London Math. Soc. (3) 13 (1963), 687-710. MR 0153743
[2] 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
[3] Corner A.L.S., Göbel R.: Prescribing endomorphism algebras, a unified treatment. Proc. London Math. Soc. (3) 50 (1985), 447-479. MR 0779399
[4] Dugas M., Göbel R.: Every cotorsion-free ring is an endomorphism ring. Proc. London Math. Soc.(3) 45 (1982), 319-336. MR 0670040
[5] Dugas M., Göbel R.: Every cotorsion-free algebra is an endomorphism algebra. Math. Z. 181 (1982), 451-470. MR 0682667
[6] 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
[7] Dugas M., Göbel R.: Torsion-free Abelian groups with prescribed finitely topologized endomorphism rings. Proc. Amer. Math. Soc. 90 (1984), 519-527. MR 0733399
[8] Eklof P., Mekler A.: On constructing indecomposable groups in $L$. J. Algebra 49 (1977), 96-103. MR 0457197 | Zbl 0372.20042
[9] Eklof P., Mekler A.: Almost Free Modules: Set-Theoretic Methods. North Holland, 1990. MR 1055083 | Zbl 1054.20037
[10] Fuchs L.: Infinite Abelian Groups. Vol. I (1970), vol. II (1973), Academic Press, New York. MR 0255673 | Zbl 0338.20063
[11] Göbel R., Goldsmith B.: Essentially indecomposable modules which are almost free. Quart. J. Math. (Oxford) (2) 39 (1988), 213-222. MR 0947502
[12] Göbel R., Goldsmith B.: Mixed modules in $L$. Rocky Mountain J. Math. 19 (1989), 1043-58. MR 1039542
[13] Göbel R., Goldsmith B.: On almost-free modules over complete discrete valuation rings. Rend. Sem. Mat. Univ. Padova 86 (1991), 75-87. MR 1154100
[14] Jech T.: Set Theory. Academic Press, New York, 1978. MR 0506523 | Zbl 1007.03002
Partner of
EuDML logo