Cotorsion-free algebras as endomorphism algebras in - the discrete and topological cases
Rüdiger E. Göbel; Brendan Goldsmith
Commentationes Mathematicae Universitatis Carolinae (1993)
- Volume: 34, Issue: 1, page 1-9
- ISSN: 0010-2628
Access Full Article
top
Abstract
topHow to cite
topGöbel, Rüdiger E., and Goldsmith, Brendan. "Cotorsion-free algebras as endomorphism algebras in $L$ - the discrete and topological cases." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 1-9. <http://eudml.org/doc/247500>.
@article{Göbel1993,
abstract = {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 constructibility, $V=L$. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived 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.},
author = {Göbel, Rüdiger E., Goldsmith, Brendan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cotorsion-free; endomorphism algebra; axiom of constructibility; Zermelo-Fraenkel set theory; realized as endomorphism algebras; torsion-free -modules; axiom of constructibility; cotorsionfree algebras; topological algebras; torsion free modules; complete discrete valuation ring},
language = {eng},
number = {1},
pages = {1-9},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cotorsion-free algebras as endomorphism algebras in $L$ - the discrete and topological cases},
url = {http://eudml.org/doc/247500},
volume = {34},
year = {1993},
}
TY - JOUR
AU - Göbel, Rüdiger E.
AU - Goldsmith, Brendan
TI - Cotorsion-free algebras as endomorphism algebras in $L$ - the discrete and topological cases
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 1
EP - 9
AB - 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 constructibility, $V=L$. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived 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.
LA - eng
KW - cotorsion-free; endomorphism algebra; axiom of constructibility; Zermelo-Fraenkel set theory; realized as endomorphism algebras; torsion-free -modules; axiom of constructibility; cotorsionfree algebras; topological algebras; torsion free modules; complete discrete valuation ring
UR - http://eudml.org/doc/247500
ER -
References
top- Corner A.L.S., Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 13 (1963), 687-710. (1963) MR0153743
- 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. MR0722629
- Corner A.L.S., Göbel R., Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), 447-479. (1985) MR0779399
- Dugas M., Göbel R., Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc.(3) 45 (1982), 319-336. (1982) MR0670040
- Dugas M., Göbel R., Every cotorsion-free algebra is an endomorphism algebra, Math. Z. 181 (1982), 451-470. (1982) MR0682667
- Dugas M., Göbel R., Almost -cyclic Abelian -groups in , in Abelian Groups and Modules (Udine 1984), CISM Courses and Lectures No. 287, Springer-Verlag, Wien-New York, 1984. MR0789809
- Dugas M., Göbel R., Torsion-free Abelian groups with prescribed finitely topologized endomorphism rings, Proc. Amer. Math. Soc. 90 (1984), 519-527. (1984) MR0733399
- Eklof P., Mekler A., On constructing indecomposable groups in , J. Algebra 49 (1977), 96-103. (1977) Zbl0372.20042MR0457197
- Eklof P., Mekler A., Almost Free Modules: Set-Theoretic Methods, North Holland, 1990. Zbl1054.20037MR1055083
- Fuchs L., Infinite Abelian Groups, Vol. I (1970), vol. II (1973), Academic Press, New York. Zbl0338.20063MR0255673
- Göbel R., Goldsmith B., Essentially indecomposable modules which are almost free, Quart. J. Math. (Oxford) (2) 39 (1988), 213-222. (1988) MR0947502
- Göbel R., Goldsmith B., Mixed modules in , Rocky Mountain J. Math. 19 (1989), 1043-58. (1989) MR1039542
- Göbel R., Goldsmith B., On almost-free modules over complete discrete valuation rings, Rend. Sem. Mat. Univ. Padova 86 (1991), 75-87. (1991) MR1154100
- Jech T., Set Theory, Academic Press, New York, 1978. Zbl1007.03002MR0506523
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.