Interpreting Modules in Modules

Abstract Rings which, from the ring-theoretic point of view, are very different may well have categories of modules which are extremely similar. More generally, the category of modules over a ring may contain many other categories of modules. Ideas from model theory are of use in elucidating this state of affairs. In particular we investigate the model-theoretic effect of tilting functors between categories of modules.

[1]  I. Reiten,et al.  Tilting in Abelian Categories and Quasitilted Algebras , 1996 .

[2]  Gregory L. Cherlin,et al.  Model Theoretic Algebra , 1976, Journal of Symbolic Logic.

[3]  K. S. Brown.  Homological criteria for finiteness , 1975 .

[4]  A. Corner,et al.  Prescribing Endomorphism Algebras, a Unified Treatment , 1985 .

[5]  C. Ringel.  Recent Advances in the Representation Theory of Finite Dimensional Algebras , 1991 .

[6]  M. Prest.  Representation embeddings and the Ziegler spectrum , 1996 .

[7]  Claus Michael Ringel,et al.  Pathological Modules over Tame Rings , 1976 .

[8]  I. Herstein,et al.  Topics in algebra , 1964 .

[9]  M. Prest.  Model theory and modules , 1988 .

[10]  Martin Ziegler,et al.  Model theory of modules , 1984, Ann. Pure Appl. Log..

[11]  L. Levy.  Modules over dedekind-like rings , 1985 .

[12]  Mike Prest,et al.  Remarks on Elementary Duality , 1993, Ann. Pure Appl. Log..

[13]  I. Herzog.  The Ziegler Spectrum of a Locally Coherent Grothendieck Category , 1997 .

[14]  I. Herzog.  Elementary duality of modules , 1993 .

[15]  S. Shelah,et al.  Kaplansky’s problem on valuation rings , 1989 .

[16]  M. Prest.  Epimorphisms of rings, interpretations of modules and strictly wild algebras , 1996 .

[17]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[18]  H. Krause.  Functors on locally finitely presented additive categories , 1998 .

[19]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[20]  C. Ringel.  Tame Algebras and Integral Quadratic Forms , 1985 .

[21]  R. Colby,et al.  Tilting, cotilting, and serially tilted rings , 1990 .

[22]  I. Assem.  Tilting theory - an introduction , 1990 .

[23]  W. Crawley-Boevey.  Tame Algebras and Generic Modules , 1991 .

[24]  C. Ringel,et al.  Representation theory of finite groups and finite-dimensional algebras : proceedings of the conference at the University of Bielefeld from May 15-17, 1991, and 7 survey articles on topics of representation theory , 1991 .

[25]  C. Ringel.  Infinite dimensional representations of finite dimensional hereditary algebras , 1979 .

[26]  I. Gelfand,et al.  COXETER FUNCTORS AND GABRIEL'S THEOREM , 1973 .

[27]  J. Trlifaj,et al.  Tilting Modules and Tilting Torsion Theories , 1995 .

[28]  Mike Prest,et al.  Model Theory of Groups and Automorphism Groups: Rings of definable scalars and biendomorphism rings , 1997 .

[29]  Martin Ziegler,et al.  Quasi finitely axiomatizable totally categorical theories , 1986, Ann. Pure Appl. Log..

[30]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[31]  A. L. S. Corner,et al.  Every Countable Reduced Torsion-Free Ring is an Endomorphism Ring , 1963 .

[32]  Anand Pillay,et al.  Introduction to stability theory , 1983, Oxford logic guides.

[33]  M. Auslander.  Large Modules over Artin Algebras , 1976 .

[34]  D. Benson,et al.  Representations and Cohomology , 1991 .

[35]  Mike Prest,et al.  Model theory and representation type of algebras , 1987 .

[36]  Helmut Lenzing.  Endlich präsentierbare Moduln , 1969 .