×

zbMATH — the first resource for mathematics

An Artin problem for division ring extensions and the pure semisimplicity conjecture. II. (English) Zbl 0953.16012
The author builds on his work in part I [Arch. Math. 66, No. 2, 114-122 (1996; Zbl 0873.16010)] which in turn extends ideas of P. M. Cohn [Proc. Lond. Math. Soc., III. Ser. 11, 531-556 (1961; Zbl 0104.03301)] and A. H. Schofield [Representation of rings over skew fields, Lond. Math. Soc. Lect. Note Ser. 92 (1985; Zbl 0571.16001)] in his efforts to find a counterexample to the pure semisimplicity conjecture (that a ring all of whose modules are direct sums of finitely generated modules must be of finite representation type).
In order to produce a counterexample to this conjecture it would be enough to produce a pair \(F\subseteq G\) of division rings and an \((F,G)\)-bimodule \(M\) such that the dimensions of the modules obtained from \(M\) by successive application of the functors \(\text{Hom}_G(_F(-)_G,G)\) and \(\text{Hom}_F(_F(-)_G,F)\) form an infinite sequence belonging to a certain set that is defined in the paper. The author shows that there are very many, indeed uncountably many, sequences any one of which would, if realized, result in the triangular matrix ring, \(R_M\), formed from \(F\), \(G\) and \(M\) being a counterexample to the conjecture. The author also obtains detailed information on the structure of the module category of any such counterexample \(R_M\).

MSC:
16G10 Representations of associative Artinian rings
16K40 Infinite-dimensional and general division rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Auslander, M., Representation theory of Artin algebras II, Comm. algebra, 1, 269-310, (1974) · Zbl 0285.16029
[2] Auslander, M., Large modules over Artin algebras, Algebra, topology and category theory, (1976), Academic Press New York, p. 3-17
[3] Auslander, M.; Reiten, I.; Smalø, S., Representation theory of Artin algebras, Cambridge studies in advanced mathematics, 36, (1995), Cambridge Univ. Press Cambridge
[4] Cohn, P.M., Quadratic extensions of skew fields, Proc. London math. soc., 11, 531-556, (1961) · Zbl 0104.03301
[5] Dowbor, P.; Ringel, C.M.; Simson, D., Hereditary Artinian rings of finite representation type, Lecture notes in math, (1980), Springer-Verlag Berlin/Heidelberg/New York/Tokyo, p. 232-241 · Zbl 0455.16013
[6] Dowbor, P.; Simson, D., A characterization of hereditary rings of finite representation type, Bull. amer. math. soc., 2, 300-302, (1980) · Zbl 0433.16023
[7] Fuller, K., On rings whose left modules are direct sums of finitely generated modules, Proc. amer. math. soc., 54, 39-44, (1976) · Zbl 0325.16024
[8] Griffith, P., On the decomposition of modules and generalized left uniserial rings, Math. ann., 184, 300-308, (1970) · Zbl 0175.31703
[9] Herzog, I., A test for finite representation type, J. pure appl. algebra, 95, 151-182, (1994) · Zbl 0814.16011
[10] Jensen, C.U.; Lenzing, H., Model theoretic algebra with particular emphasis on fields, rings, modules, Algebra, logic and applications, 2, (1989), Gordon & Breach New York · Zbl 0728.03026
[11] Kelly, G.M., On the radical of a category, J. austral. math. soc., 4, 299-307, (1964) · Zbl 0124.01501
[12] Köthe, G., Verallgemeinerte abelsche gruppen mit hyperkomplexen operatorenring, Math. Z., 39, 31-44, (1934) · JFM 60.0102.01
[13] Krause, H., Dualizing rings and a characterization of finite representation type, C. R. acad. sci. Paris, Sér. I, math., 322, 507-510, (1996) · Zbl 0852.16008
[14] Ringel, C.M., Representations of K-species and bimodules, J. algebra, 41, 269-302, (1976) · Zbl 0338.16011
[15] Ringel, C.M.; Tachikawa, H., QF-3 rings, J. reine angew. math., 272, 49-72, (1975) · Zbl 0318.16006
[16] Schofield, A.H., Representations of rings over skew fields, London math. soc. lecture notes series, 92, (1985), Cambridge Univ. Press Cambridge
[17] Schmidmeier, M., The local duality for homomorphisms and an application to pure semisimple PI-rings, Colloq. math., 77, 121-132, (1998) · Zbl 0915.16001
[18] Simson, D., Functor categories in which every flat object is projective, Bull. polon. acad. sci., ser. math., 22, 375-380, (1974) · Zbl 0328.18005
[19] Simson, D., Pure semisimple categories and rings of finite representation type, J. algebra, 48, 290-296, (1977) · Zbl 0409.16030
[20] Simson, D., Partial Coxeter functors and right pure semisimple hereditary rings, J. algebra, 71, 195-218, (1981) · Zbl 0477.16014
[21] Simson, D., Linear representations of partially ordered sets and vector space categories, Algebra, logic and applications, 4, (1992), Gordon & Breach
[22] Simson, D., On right pure semisimple hereditary rings and an Artin problem, J. pure appl. algebra, 104, 313-332, (1995) · Zbl 0848.16013
[23] Simson, D., An Artin problem for division ring extensions and the pure semisimplicity conjecture, I, Arch. math., 66, 114-122, (1996) · Zbl 0873.16010
[24] Simson, D., A class of potential counter-examples to the pure semisimplicity conjecture, (), 345-373 · Zbl 0936.16010
[25] Simson, D., Dualities and pure semisimple rings, Proceedings of the conference abelian groups, module theory and topology, university of Padova, June 1997, Lecture notes in pure and appl. math., (1998), Dekker, p. 381-388 · Zbl 0926.16004
[26] Simson, D., On the representation theory of artinian rings and Artin’s problems on division ring extensions, Bull. Greek math. soc., 42, 97-112, (1999) · Zbl 0988.16009
[27] Simson, D.; Skowroński, A., The Jacobson radical power series of module categories and the representation type, Bol. soc. mat. mexicana, 5, 223-236, (1999) · Zbl 0960.16012
[28] Zimmermann, W., Einige charakterisierung der ringe über denen reine untermoduln direkte summanden sind, Bayer. akad. wiss. math.-natur., abt., II, 77-79, (1973) · Zbl 0276.16023
[29] Zimmermann-Huisgen, B.; Zimmermann, W., On the sparsity of representations of rings of pure global dimension zero, Trans. amer. math. soc., 320, 695-711, (1990) · Zbl 0699.16019
[30] Zimmermann-Huisgen, B., Strong preinjective partitions and representation type of Artinian rings, Proc. amer. math. soc., 109, 309-322, (1990) · Zbl 0703.16011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.