Structure Theorems for Pure Semisimple Grothendieck Locally PI-Categories

References

• Milnor , J. and Moore , J. C. 1965 . On the Structure of Hopf Algebras . Annals of Math , 81 : 211 – 264 .
   Web of Science ®Google Scholar
• Gabriel , P. 1960 . Sur les catégories Abéliennes Localement Noetheriennes et leurs applications aux Algbres Tudiées par Dieudonné Seminaire Serre; College de France: Paris
   Google Scholar
• Schoeller , C. 1970 . Etude De La categorie Des Algbres De Hopf Commutatives Connexes Sur Un Corps . Manuscripta Math , 3 : 133 – 155 .
   Google Scholar
• Simson , D. and Skowroński , A. 1978 . On the Category of Commutative Connected Graded Hopf Algebras over a Perfect Field . Fund. Math. , 101 : 137 – 149 .
   Google Scholar
• Skowroński , A. 1980 . The Category of Abelian Hopf Algebras . Fund. Math. , 107 : 167 – 193 .
   Google Scholar
• Skowronski , A. 1983 . Projective Abelian Hopf Algebras over a Field . Dissertationes Math. PWN Warszawa , 205 : 1 – 42 .
   Google Scholar
• Chin , W. and Montgomery , S. 1997 . Basic coalgebras . AMS/IP Studies in Advanced Mathematics , 4 : 41 – 47 .
   Google Scholar
• Green , J. A. 1976 . Locally Finite Representations . J. Algebra , 41 : 137 – 171 .
   Web of Science ®Google Scholar
• García , J. M. , Jara , P. and Merino , L. M. 1997 . Decomposition of Locally Finite Modules Preprint
   Google Scholar
• Gómez , Torrecillas J. , Jara , P. and Merino , L. 1996 . Locally Finite Representations of Algebras . Comm. Algebra , 24 : 4581 – 4601 .
   Google Scholar
• Castano , Iglesias F. and Gómez , Torrecillas J. 1998 . Wide Morita Context and Equivalences of Comodule Categories . J. Pure Appl. Algebra , 131 : 213 – 225 .
   Google Scholar
• Nastasescu , C. , Torrecillas , B. and Zhang , Y. H. 1996 . Hereditary Coalgebras . Comm. Algebra , 24 : 1521 – 1528 . 299 – 307 .
   Web of Science ®Google Scholar
• Simson , D. 1977 . On Pure Global Dimension of Locally Finitely Presented Grothendieck Categories . Fund. Math. , 96 : 91 – 16 .
   Google Scholar
• Simson , D. 1982 . On the Structure of Locally Finite Pure Semisimple Grothendieck Categories . Cahiers de Topologie et Geom. Diff. , 33 : 397 – 406 .
   Google Scholar
• Sweedler , M. E. 1969 . Hopf Algebras , New York : Benjamin .
   Google Scholar
• Woodcock , D. 1977 . Some Categorical Remarks on the Representation Theory of Coalgebras . Comm. Algebra , 25 : 2775 – 2794 .
   Google Scholar
• Nowak , S. and Simson , D. 2001 . Locally Dynkin Quivers and Hereditary Coalgebras Whose Left Comodules are Direct Sums of Finite Dimensional Comodules . Comm. Algebra , 29 in press
   Google Scholar
• Simson , D. 2001 . Coalgebras, Comodules, Pseudocompact Algebras and tame Comodule Type . Colloq. Math. , in press
   Google Scholar
• Albu , T. and Nastasescu , C. 1982 . Relative Finiteness in Module Theory , New York : Marcel Dekker .
   Google Scholar
• Azumaya , G. 1992 . Countable Generatedness Version of Rings of Pure Global Dimension Zero , London Math. Soc. Lecture Notes Series Vol. 168 , 43 – 79 . Cambridge University Press .
   Google Scholar
• Azumaya , G. and Facchini , A. 1989 . Rings of Pure Global Dimension Zero and Mittag-Leffler Modules . J. Pure Appl. Algebra , 62 : 109 – 122 .
   Web of Science ®Google Scholar
• Brune , H. 1979 . Some Left Pure Semisimple Ringoids which are not Right Pure Semisimple . Comm. Algebra , 7 : 1795 – 1803 .
   Web of Science ®Google Scholar
• Dung , N. V. 1997 . Modules with Indecomposable Decompositions that Complement Maximal Direct Summands . J. Algebra , 197 : 449 – 467 .
   Google Scholar
• Dung , N. V. 1998 . Indecomposable Decompositions of Pure-Injective Modules . Comm. Algebra , 26 : 3709 – 3725 .
   Google Scholar
• Facchini , A. 1989 . Aneli di Tipo di Rappresentazione Finita, di Dimensione Pura Globale Zero, e Moduli di Mittag-Leffler . Rendiconti Sem. Mat. Fiz. Milano , 59 : 65 – 80 .
   Google Scholar
• Facchini , A. 1991 . Mittag-Leffler Modules, Reduced Products and Direct Products . Rend. Sem. Mat. Univ. Padova , 85 : 119 – 132 .
   Google Scholar
• Gómez , Pardo J.L. and Guil , Asensio P. A. 1997 . Indecomposable Decompositions of Finitely Presented Pure-Injective Modules . J. Algebra , 192 : 200 – 208 .
   Google Scholar
• Menini , C. 1988 . “ Gabriel–Popescu Type Theorems and Graded Modules ” . In Perspective in Ring Theory (Antwerp, 1987) , 239 – 251 . Dordrecht : Kluwer Acad. Publ. .
   Google Scholar
• Wisbauer , R. 1988 . Foundations of Module and Ring Theory. Algebra, Logic and Applications , Vol. 3 , Gordon & Breach Science Publishers .
   Google Scholar
• Simson , D. 1977 . Pure Semisimple Categories and Rings of Finite Representation Type . J. Algebra , 48 : 290 – 296 . Corrigendum: J. Algebra 1980, 67: 254–256
   Web of Science ®Google Scholar
• Auslander , M. 1974 . Representation Theory of Artin Algebras II . Comm. Algebra , 1 : 269 – 310 .
   Google Scholar
• Fuller , K. 1976 . On Rings whose Left Modules are Direct Sums of Finitely Generated Modules . Proc. Amer. Math. Soc. , 54 : 39 – 44 .
   Web of Science ®Google Scholar
• Ringel , C. M. and Tachikawa , H. 1975 . QF-3 rings . J. Reine Angew. Math. , 272 : 49 – 72 .
   Web of Science ®Google Scholar
• Simson , D. 1974 . Functor Categories in which Every Flat Object is Projective . Bull. Pol. Acad. Sci.: Math. , 22 : 375 – 380 .
   Google Scholar
• Zimmermann-Huisgen , B. 1979 . Rings whose Right Modules are Direct Sums of Indecomposable Modules . Proc. Amer. Math. Soc. , 77 : 191 – 197 .
   Web of Science ®Google Scholar
• Zimmermann , W. 1973 . Einige Charakterisierung der Ringe über Denen Reine Untermoduln Direkte Summanden Sind . Bayer. Akad. Wiss. Math.-Natur. , : 77 – 79 . Abt. II
   Google Scholar
• Zimmermann-Huisgen , B. and Zimmermann , W. 1990 . On the Sparsity of Representations of Rings of Pure Global Dimension Zero . Trans. Amer. Math. Soc. , 320 : 695 – 711 .
   Web of Science ®Google Scholar
• Simson , D. 1995 . On Right Pure Semisimple Hereditary Rings and an Artin Problem . J. Pure Appl. Algebra , 104 : 313 – 332 .
   Web of Science ®Google Scholar
• Simson , D. 1996 . An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture . I. Archiv fur Math. , 66 : 114 – 122 .
   Web of Science ®Google Scholar
• Simson , D. 1997 . “ A Class of Potential Counter-Examples to the Pure Semisimplicity Conjecture ” . In Advances in Algebra and Model Theory , Algebra, Logic and Applications Series Edited by: Droste , M. and Göbel , R. Vol. 9 , 345 – 373 . Australia : Gordon & Breach Science Publishers .
   Google Scholar
• Herzog , I. 1994 . A Test for Finite Representation Type . J. Pure Appl. Algebra , 95 : 151 – 182 .
   Web of Science ®Google Scholar
• Dung , N. V. 1999 . Preinjective Modules and Finite Representation type of Artinian Rings . Comm. Algebra , 27 : 3921 – 3947 .
   Web of Science ®Google Scholar
• Jensen , C. U. and Lenzing , H. 1989 . Model Theoretic Algebra With Particular Emphasis on Fields, Rings, Modules; Algebra, Logic and Applications , Vol. 2 , Gordon & Breach Science Publishers .
   Google Scholar
• Fox , T. F. 1976 . Purity in Locally-Presentable Monoidal Categories . J. Pure Appl. Algebra , 8 : 1261 – 1265 .
   Google Scholar
• Simson , D. October 1994 . Remarks on Pure Semisimple Categories October , Manuscript notes. Murcia and April 1997
   Google Scholar
• Simson , D. 1978 . On Pure Semi-Simple Grothendieck Categories . I. Fund. Math. , 100 : 211 – 222 .
   Google Scholar
• Simson , D. 1980 . On Pure Semi-Simple Grothendieck Categories. II . Fund. Math. , 110 : 107 – 116 .
   Google Scholar
• Popescu , N. 1973 . Abelian Categories With Applications to Rings and Modules , Academic Press .
   Google Scholar
• Stenström , B. 1975 . Rings of Quotients , Springer-Verlag .
   Google Scholar
• Simson , D. 1981 . Partial Coxeter Functors and Right Pure Semisimple Hereditary Rings . J. Algebra , 71 : 195 – 218 .
   Web of Science ®Google Scholar
• Gabriel , P. 1962 . Des Catégorie Abéliennes . Bull. Soc. Math. France , 90 : 323 – 448 .
   Google Scholar
• García , J. L. and Martínez Hernández , J. 1993 . Purity through Gabriel's Functor Rings . Bull. Soc. Math. Belgique , 45 : 137 – 152 .
   Google Scholar
• García , J. L. and Simson , D. 1997 . On Rings whose Flat Modules form a Grothendieck Category . Coll. Math. , 73 : 115 – 141 .
   Google Scholar
• Simson , D. 1992 . Linear Representations of Partially Ordered Sets and Vector Space Categories; Algebra, Logic and Applications , Vol. 4 , Gordon & Breach Science Publishers .
   Google Scholar
• Rosenberg , A. and Zelinsky , D. 1959 . Finiteness of the Injective Hull . Math. Z. , 70 : 372 – 380 .
   Google Scholar
• Anderson , F. W. and Fuller , K. R. 1991 . Rings and Categories of Modules; , 2nd Ed. , Springer-Verlag .
   Google Scholar
• Freyd , P. 1964 . Abelian Categories , New York : Harper and Row .
   Google Scholar
• García Hernández , J. L. , Gómez Pardo , J. L. and Martínez Hernández , J. 1986 . Semiperfect Modules with Respect to a Torsion Theory . J. Pure and Appl. Algebra , 43 : 145 – 172 .
   Web of Science ®Google Scholar
• Tachikawa , H. 1973 . Quasi-Frobenius Rings and Generalizations , Vol. 351 , Berlin-Heidelberg, New York : Springer-Verlag . Lecture Notes in Math.
   Google Scholar
• Drozdowski , G. and Simson , D. 1979 . Quivers of Pure Semisimple Type . Bull. Pol. Acad. Sci. Math. , 27 : 33 – 40 .
   Google Scholar
• Gabriel , P. 1972 . Unzerlegbare Darstellungen, I . Manuscripta Math , 6 : 71 – 103 .
   Web of Science ®Google Scholar
• Simson , D. and Skowronski , A. 1999 . The Jacobson Radical Power Series of Module Categories and the Representation Type . Bol. Soc. Mat. Mexicana , 5 : 223 – 236 .
   Web of Science ®Google Scholar
• Dung , N. V. and García , J. L. 2001 . Categories of Locally Finite Representation Type . J. Algebra , 238 : 200 – 238 .
   Web of Science ®Google Scholar