Model Theory and Modules, London Mathematical Society Lecture Notes Series Vol. 130, Cambridge University Press, Cambridge, 1988 (400pp).
(with J. Humphreys) Numbers, Groups and Codes, Cambridge University Press, 1989 (288pp). A second, considerably revised edition was published in 2004 An extensive list of corrections is on this webpage.
Purity, Spectra and Localisation, Encyclopedia of Mathematics
and its Applications, Vol. 121, Cambridge University Press, 2009 (xxviii+769pp).
Typos, corrections
and comments
Links are to drafts of papers which, in some cases, may differ somewhat from the published version.
1. Some model-theoretic aspects of torsion theories, J. Pure Applied Algebra, 12(3) (1978), 295-310.
2. Torsion and universal Horn classes of modules, J. London Math. Soc., 19(3) (1979), 411-416.
3. Model-completions of some theories of modules, J. London Math. Soc., 20(3) (1979), 369-372.
4. Elementary torsion theories and locally finitely presented Abelian categories, J. Pure Applied Algebra, 18(2) (1980), 205-212.
5. Quantifier elimination for modules, pp. 109-129 in "Proceedings of the Model Theory Meeting at Brussels and Mons 1980" in Bull. Soc. Math. Belgique, 33 (1981).
6. Elementary equivalence of sigma-injective modules, Proc. London Math. Soc., 45(1) (1982), 71-88.
7. Existentially complete prime rings, J. London Math. Soc., 28(2) (1983), 238-246.
8. (with A. Pillay) Forking and pushouts for modules, Proc. London Math. Soc., 46(2) (1983), 365-384.
9. Rings of finite representation type and modules of finite Morley rank, J. Algebra, 88(2) (1984), 502-533.
10. The generalised RK-order, orthogonality and regular types for modules, J. Symbolic Logic, 50(1) (1985), 202-219.
11. (with A. Pillay) Modules and stability theory, Trans. Amer. Math. Soc., 300(2) (1987), 641-662.
12. Model theory and representation type of algebras, pp. 219-260 in Logic Colloquium '86 (Proceedings of the European meeting of the Association for Symbolic Logic, Hull, 1986), North-Holland, Amsterdam, 1988.
13. (with F. Point) Decidability for theories of modules, J. London Math. Soc. 38(2) (1988), 193-206.
14. Duality and pure-semisimple rings, J. London Math. Soc. 38 (1988), 403-409.
15. (with A. Marcja and C. Toffalori) Classification theory for abelian groups with an endomorphism, Arch. Math. Logic, 31(2) (1991), 95-104.
16. Wild representation type and undecidability, Communications in Algebra, 19(3) (1991), 919-929.
17. (with T. Kucera) Imaginary modules, J. Symbolic Logic, 57(2) (1992), 698-723.
18. (with T. Kucera) Four concepts from ``geometrical'' stability theory in modules, J. Symbolic Logic, 57(2) (1992), 723-740.
19. (with A. Marcja and C. Toffalori) The stability classification for abelian-by-finite groups and modules over group rings, J. London Math. Soc., 47(2) (1993), 212-226.
20. (with A. Marcja and C. Toffalori) On wildness of abelian-by-finite groups, Ann. Pure Applied Logic, 62(2) (1993), 167-173.
21. Remarks on elementary duality, Ann. Pure Applied Logic, 62(2) (1993), 183-205.
22. (with Ph. Rothmaler and M. Ziegler) Extensions of elementary duality, J. Pure Applied Algebra, 93(1) (1994), 33-56.
23. (with Ph. Rothmaler and M. Ziegler) Absolutely pure modules and indiscrete rings, J. Algebra, 174(2) (1995), 349-372.
24. Epimorphisms of rings, interpretations of modules and strictly wild algebras, Comm. Algebra, 24(2) (1996), 517-531.
25. Representation embeddings and the Ziegler spectrum, J. Pure Applied Algebra, 113(3) (1996), 315-323.
26. (with S. Baratella) Pure-injective modules over the dihedral algebras, Comm. Algebra, 25(1) (1997), 11-32.
27. Interpreting modules in modules, Ann. Pure Applied Logic, 88(2-3) (1997), 193-215.
28. (with K. Burke) Rings of definable scalars and biendomorphism rings, pp.188-201 in Model Theory of Groups and Automorphisms, London Math. Soc. Lecture Note Ser. Vol. 244, 1997.
29. Morphisms between finitely presented modules and infinite-dimensional representations, Canad. Math. Soc. Conf. Proc. Ser., Vol. 24 (1998), 447-455.
30. The representation theories of elementarily equivalent rings, J. Symbolic Logic, 63(2) (1998), 439-450.
31. Ziegler spectra of tame hereditary algebras, J. Algebra, 207(1) (1998), 146-164.
32. The sheaf of locally definable scalars over a ring, pp. 339-351 in Models and Computability, London Math. Soc. Lecture Note Ser., Vol. 259, Cambridge University Press, 1999.
33. (with G. Puninski and Ph. Rothmaler) Rings described by various purities, Comm. Algebra, 27(5) (1999), 2127-2162.
34. Tensor product and theories of modules, J. Symbolic Logic, 64(2) (1999), 617-628.
35. (with V. Puninskaya) Vaught's conjecture for modules over a commutative Prüfer ring, Algebra i Logika, 38(4) (1999), 419-435.
36. (with V. Puninskaya) Modules with few types over a commutative Prüfer ring, (in Russian), Uspekhi Mat. Nauk, 54(3) (1999), 149-150.
37. (with G. Puninski) \Sigma-pure-injective modules over a commutative Prüfer ring, Comm. Algebra, 27(2) (1999), 961-971.
38. (with G. Puninski) Some model theory over hereditary noetherian domains, J. Algebra, 211(1) (1999), 268-297.
39. Topological and geometric aspects of the Ziegler spectrum, in Proceedings of Conference on Infinite Length Modules, Bielefeld, 1998, Birkhäuser, 2000.
40. (with K. Burke) The Ziegler and Zariski spectra of some domestic string algebras, Algebras and Representation Theory, 5(3) (2002), 211-234. .pdf
41. (with V. Puninskaya) Modules with few types over some finite-dimensional algebras, J. Symbolic Logic, 67(2) (2002), 841-858.
42. (with J. Schröer) Serial functors, Jacobson radical and representation type, J. Pure Applied Algebra, 170(2-3) (2002), 295-307/
43. (with A. Marcja and C. Toffalori) The torsionfree part of the Ziegler spectrum of $RG$ where $R$ is a Dedekind domain and $G$ is a finite group, J. Symbolic Logic, 67(3) (2002), 1126-1140.
44. (with G. Puninski) Pure injective envelopes of finite length modules over a Generalised Weyl Algebra, J. Algebra, 251(1) (2002), 150-177. .pdf
45. Model theory for algebra, pp. 199-226 in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 3, Elsevier, 2003. .pdf
46. Model theory and modules, pp. 227-253 in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 3, Elsevier, 2003. .pdf
47. (with S. Crivei and G. Reynders) Model theory of comodules, J. Symbolic Logic, 69(1) (2004), 137-142. .pdf
48. (with R. Wisbauer) Finite presentation and purity in categories sigma[M], Colloq. Math., 99(2) (2004), 189-202. .pdf
49. (with V. Puninskaya and A. Ralph) Model theory of sheaves of modules, J. Symbolic Logic, 69(4) (2004), 1187-1200. .pdf
50. (with G. Garkusha) Injective objects in triangulated categories, J. Algebra Applications, 3(4) (2004), 367-389. .pdf
51. (with G. Puninski) One-directed indecomposable pure-injective modules over string algebras, Colloq. Math., 101(1) (2004), 89-112. .ps
52. Ideals of mod-R and the omega-radical, J. London Math. Soc., 71(2) (2005), 321-334. .pdf
53. (with G. Garkusha) Triangulated categories and the Ziegler spectrum, Algebras and Representation Theory, 8(4) (2005), 499-523. .pdf
54. (with G. Puninski) Krull-Gabriel dimension of 1-domestic string algebras, Algebras and Representation Theory, 9(4) (2006), 337-358. .pdf
55. (with S. L'Innocente) Rings of definable scalars of Verma modules, J. Algebra Apps., 6(5) (2007), 779-787 .pdf
56. (with G. Garkusha) Classifying Serre subcategories of finitely presented modules, Proc. Amer. Math. Soc., 136(3) (2008), 761-770. doi.org/10.1090/S0002-9939-07-08844-2 .pdf
57. (with G. Garkusha) Reconstructing projective schemes from Serre subcategories, J. Algebra, 319(2) (2008), 1132-1153. doi.org/10.1016/j.algebra.2007.10.034 .pdf
58. (with G. Garkusha) Torsion classes of finite type and spectra, pp. 393-412 in K-theory and Noncommutative Geometry, European Math. Soc., 2008. DOI:10.4171/060-1/12 .pdf
59. (with R. Rajani) Model-theoretic imaginaries and coherent sheaves, Applied Categorical Structures, 17(6) (2009), 517-559. doi.org/10.1007/s10485-008-9151-6 .pdf.
60. (with R. Rajani) Pure-injectivity and model theory for G-sets, J. Symbolic Logic, 74(2) (2009), 474-488. doi.org/10.2178/jsl/1243948323 .pdf
61. Definable additive categories, in S. Breaz, S. Crivei and A. Marcus (Eds.), Proceedings of the International Conference on Modules and Representation Theory (Cluj-Napoca, July 7-12, 2008), Cluj University Press, Cluj-Napoca, 2009 .pdf
62. Pure-injective modules, AJSE-Mathematics, 1(1) (2009), 175-191 .pdf
63. (with S. Crivei and B. Torrecillas) Covers in finitely accessible categories, Proc. Amer. Math. Soc., 138 (2010), 1213-1221. doi.org/10.1090/S0002-9939-09-10178-8 .pdf
64. (with R. Rajani) Structure sheaves of definable additive categories, J. Pure Applied Algebra, 214 (2010), 1370-1383. DOI:10.1016/j.jpaa.2009.11.003 .pdf
65. Definable additive categories: purity and model theory, Mem. Amer. Math. Soc., Vol. 210, 2011. DOI:10.1090/S0065-9266-2010-00593-3 .pdf
66. Model theory in additive categories, in "Models, Logics and Higher-Dimensional Categories: a Tribute to the Work of Mihaly Makkai", CRM Proceedings and Lecture Notes, Vol. 53, Amer. Math. Soc., 2011.
67. (with S. Estrada, P. A. Guil Asensio and J. Trlifaj) Model category structures arising from Drinfeld vector bundles, Adv. in Math., 231(3-4) (2012), 1417-1438. DOI:10.1016/j.aim.2012.06.011 .pdf
68. (with H. Krause) The Gabriel-Roiter filtration of the Ziegler spectrum, Quart. J. Math., 64(3) (2013), 891-901. doi.org/10.1093/qmath/has020 .pdf
69. Superdecomposable pure-injective modules, pp. 263-296 in Advances in Representation Theory of Algebras (ICRA Bielefeld 2012), European Math. Soc., 2014 .pdf
70. (with A. R. Mehdi) Almost dual pairs and definable classes of modules, Comm. Algebra, 43(4) (2015), 1387-1397. DOI:10.1080/00927872.2013.865048 arXiv
71. (with R. Harland) Modules with irrational slope over tubular algebras, Proc. London Math. Soc., 110(3) (2015), 695-720. DOI:10.1112/plms/pdu068 on arXiv
72. (with G. Puninski) Ringel's conjecture for domestic string algebras, Math. Zeit., 282(1) (2016), 61-77, DOI:10.1007/s00209-015-1532-6. on arXiv
73. (with L. Gregory) Representation embeddings, interpretation functors and controlled wild algebras, J. London Math. Soc., 94(3) (2016), 747-766, DOI:10.1112/jlms/jdw055. on arXiv
74. (with K. K. Arnesen, R. Laking and D. Pauksztello) The Ziegler spectrum for derived-discrete algebras, Adv. in Math., 319 (2017), 653-698. DOI:10.1016/j.aim.2017.07.016 on arXiv
75. (with L. Barbieri-Viale) Definable categories and T-motives, Rendiconti Sem. Mat. Univ. Padova, 139 (2018), 205-224. DOI:10.4171/RSMUP/139-8 on arXiv
76. (with R. Laking and G. Puninski) Krull-Gabriel dimension of domestic string algebras, Trans. Amer. Math. Soc., 370 (2018), 4813-4840. DOI:10.1090/tran/7093 on arXiv
77. Modules as exact functors, pp. 37-65 in Surveys in Representation Theory of Algebras, Contemporary Mathematics, Vol. 716, Amer. Math. Soc., 2018. DOI:10.1090/conm/716 on arXiv
78. Multisorted modules and their model theory, pp. 115-151 in Model Theory of Modules, Algebras and Categories, Contemporary Mathematics, Vol. 730, Amer. Math.Soc., 2019. DOI:10.1090/conm/730 on arXiv
79. Model theory for sheaves of modules, pp. 89-102 in Logic and its Applications, Proc. 8th Indian Conf. ICLA, Delhi, Lecture Notes in Comp. Sci, Vol. 11600, Springer, 2019. DOI:10.1007/978-3-662-58771-3_9
80. (with L. Barbieri-Viale) Tensor product of motives via Kunneth formula, J. Pure Applied Algebra, 224(6) (2020). DOI:10.1016/j.jpaa.2019.106267 on arXiv
81. (with L. Barbieri-Viale and A. Huber) Tensor structure for Nori motives, Pacific J. Math., 306(1) (2020), 1-30. DOI:10.2140/pjm.2020.306-1 on arXiv
82. (with A. Slavik) Purity in categories of sheaves, Math. Zeit., 297(1-2) (2021), 429-451. DOI:10.1007/s00209-020-02517-5 on arXiv
83. Definable categories and monoidal categories, pp. 197-222 in Proceedings of Conference on Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods, Almeria, July 2022, Springer, 2024 .pdf
84. (with R. Wagstaffe) Model theory in compactly generated (tensor-)triangulated categories, Model Theory, 3(1) (2024), 147-197 on arXiv
85. Strictly atomic modules in definable categories, Annals of Representation Theory, 1(2) (2024), 299-334 on arXiv
The Zariski spectrum of the category of finitely presented modules, preprint, 1998, revised 2004, 2006 and 2008. .pdf
(with A. Ralph) Locally finitely presented categories of sheaves of modules. .ps, now split into a shortened paper (revised 21/09/18) of the same title .pdf and On sheafification of modules, .pdf, an exposition of some background.
(with R. Rajani) Positive imaginaries and coherent affine functors .pdf
Abelian categories and definable additive categories on arXiv
Categories of imaginaries for definable additive categories on arXiv
Spectra of small abelian categories on arXiv
Model Theory of Modules .pdf, prepared (2020) for a volume of "Research Trends in Contemporary Logic"
Tensor and direct extension of definable subcategories on arXiv
Notes on Model Theory; notes from lectures on intermediate model theory from the MODNET Summer School held at Leeds, December 2005.
Spectra of Categories of Modules
Sheaves of Rings of Definable Scalars ; Conference in honour of Helmut Lenzing, Paderborn, 2006
Model Theory in Additive Categories; Model Theory conference, Seoul, 2010
Categories of Imaginaries for Additive Structures
The Functor Category Notes for lectures given at the conference in Bristol on Categorical Methods in Representation Theory, September 2012.
Summer School and Workshop on Infinite-dimensional Representations of Finite-dimensional Algebras, Manchester, September 2015
Model Theory and Modules Graduate course, Granada, November 2015
Model-theoretic Imaginaries and Localisation for Additive Categories; Topos Theory conference, IHES, Paris, 2015
The Compactness Theorem, Notes and Beamer slides, London Mathematical Society Undergraduate Summer School, Manchester 2017.
Model Theory, Notes for an undergraduate course in model theory.
Conference in memory of Gena Puninski, Manchester, 6-9th April, 2018.
Definability in Module Theory, Isfahan School and Conference on Representations of Algebras, Isfahan, April 2019.
Lorna Gregory
David Pauksztello
Rosanna Laking
Sarah Mattingly: On the Structure of Modules of Finite Rank (1994)
Kevin Burke: Some Model-theoretic Properties of Functor Categories for Modules (1994)
Robert Combley: Ordering the Formulae of a Stable Theory by Representability (1995)
Shahabaddin Ebrahimi Atani On Indecomposable Pure-Injective Modules over Pullback Rings (1996)
Geert Reynders: Ziegler Spectra over Serial Rings and Coalgebras (1998) .ps
Nicolas Granger: Stability, Simplicity and the Model Theory of Bilinear Forms (1999) .ps
Alexandra Ralph: An Approach to the Model Theory of Sheaves of Modules (2001)
Ravi Rajani: Model-Theoretic Imaginaries and Ziegler Spectra in General Categories (2007) .ps
Richard Harland: Pure-injective Modules over Tubular Algebras and String Algebras (2011).pdf
Lorna Gregory: Ziegler Spectra of Valuation Rings (2011).pdf
Simon Perera: Grothendieck Rings of Theories of Modules (2011).pdf
Philip Bridge: Essentially Algebraic Theories and Localizations in Toposes and Abelian Categories (2012) .pdf
Akeel Ramadan Mehdi: Purity Relative to Classes of Finitely Presented Modules (2013) .pdf
Amit Kuber: K-Theory of Theories of Modules and Algebraic Varieties (2014) .pdf
Rosanna Laking: String Algebras in Representation Theory (2016) .pdf
Samuel Dean: Dualities and Finitely Presented Functors (2017) .pdf
Michael Bushell: Ziegler Spectra for Self-Injective Algebras of Polynomial Growth (2018) .pdf
Harry Gulliver: The Injective Spectrum of a Right Noetherian Ring (2019) .pdf
Isaac Bird: Definable Subcategories of Maximal Cohen-Macaulay Modules (2019) .pdf
Zoe Gosling (co-supervised with Prof. Daniela Caselli): Mathematics and Modernism (2020)
Rose Wagstaffe: Definability in Monoidal Additive and Tensor Triangulated Categories (2021) original pdf with a correction, corrected pdf
Soinbhe Nic Dhonncha: Purity and Chains of Modules (2024) .pdf
Nicholas Mosley: The Problem of Quantifier-Elimination for the Real Number Field with Exponential Function (1989)
John Smith: On Connections between Category Theory and Logic (1994)
Tristram de Piro: The Model Theory of Lie Algebras (1996)
Alexandra Ralph: An Approach to the Model Theory of Sheaves (1996)
Cecily Crampin: Models as Functors for T^{eq} (1997)
Timothy Mellor: Elimination of Quantifiers for Modules over the Path Algebra \tilde{A_1}(k) (1997)
Julie Fowler: Induction, Restriction and Elementary equivalence (1998)
James Gray: On Definable Sets over Finite Fields (1998)
Owain Derrick: Model Theory of Pseudofinite Fields (1999)
Ralph Sherrington: Quantum Computation (2001)
Martin Carpenter: Model Theory and Grothendieck Topologies (2001)
Owen Jones: Elimination of Quantifiers for Modules (2001)
Mahmood Sohrabi: Localization in Abelian and Accessible Categories (2002)
Jonathan Roberts: Model Theory of Fields (2005)
Ray-Ming Chen: O-Minimality (2006)
John Bourke: Interpretations and Automorphism Groups (2006)
David Crowther: Grothendieck Rings of Theories (2006)
Victoria Fellows: Set Theory (2006)
Andrew Bannatyne: The Whitehead Problem (2006)
François Wirion: Presenting Theories in Category Theory (2008)
Vladimir Lukiyanov: Well Generated Triangulated Categories (2010)
Vladimir Lukiyanov: Endofinite Modules (MPhil) (2011)
Connor Ford: Indecomposable Pure-Injective Modules over String Algebras (2018)
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