Department of Mathematics, Faculty of Mathematics and Computer Science "Babes-Bolyai" University, Cluj-Napoca

Research Project
"Functors on Module Categories",
October 2011-September 2014.

code: PN-II-RU-TE-2011-3-0065; contract: 47/05.11.2011

Financed by
Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii , UEFISCDI

Team:

Simion Breaz (CV) ,
George Ciprian Modoi (CV),
Septimiu Crivei(CV)
Cosmin Pelea(CV),
Flaviu Pop (CV).

Claudiu Valculescu (CV) (former colleague).

Abstract:

(I) Scientific objectives

This project is focused on two sorts of objectives, as follows:
A) SCIENTIFIC OBJECTIVES: The presumed research activity is developed on the following directions:
(I) FUNCTORS ON MODULE CATEGORIES: (a) Commuting, respectively semi-commuting, properties (as those used for small and Mittag-Leffler modules) of functors defined on module categories or on other associated categories: quotient, stable or derived categories. (b) Functors connecting module categories with other categories (e.g. connections with topological spaces via spectra). (c) Equivalences and dualities between module categories (e.g. Morita and natural equivalences and dualities, Warfield dualities, Arnold-Fomin-Wickless duality).
(II) TORSION AND COTORSION THEORIES; PERPENDICULAR CLASSES. (a) Torsion theories and kernels of functors (the interplay between the structure of an objects and properties of the kernel of some related functors); (b) Cotorsion theories and cotorsion triples, approximations of modules and construction of adjoint functors; (c) Closure properties for classes of static, adstatic or reflexive modules, Baer splitting properties, connections with the structure of projective modules.
1.Completion of the research infrastructure by assuring the bibliographical background (books, journals) and the material background of the research (computers, office consumables etc.) 2.Improving the level of knowledge of the team members and other young researchers and students.

Achievements:
Report 2011
Report 2012
Report 2013
Report 2014
(A) Papers in ISI journals

1. U. Albrecht, S. Breaz, P. Schultz, The Ext functor and self-sums, to appear in Forum Math., Abstract , DOI: 10.1515/forum-2011-0141
2. S. Crivei, H. Inankil, M.T. Kosan, G. Olteanu, Correspondences of coclosed submodules, to appear in Communications in Algebra., 41 (10), 3635-3647 (2013) Abstract
3. C.G. Modoi, A representability theorem for some huge abelian categories, Homology, Homotopy and Applications, Volume 14, Number 2 (2012), 23-36. Abstract
4. S. Breaz , Modules M such that Ext^1(M,-) commutes with direct limits, to appear in Algebras and Representation Theory, Abstract
5. C. Pelea, Hyperrings and $\alpha^*$-relations. A general approach, Journal of Algebra , 383 (2013), 104-128. Abstract
6. U. Albrecht, S. Breaz: A note on self-small modules over RM-domains, Journal of Algebra and its Applications, vol. 13 no. 1 (2014), 1350073 (8pages) Abstract
7. S. Crivei, Essential and retractable Galois connections, Journal of Algebra and its Applications, Vol. 12, No. 06 (2013) 1350017 Abstract
8. G. C. Modoi, The dual of Brown representability for homotopy categories of complexes, Journal of Algebra, Volume 392, 15 October 2013, Pages 115–124 Abstract
9. S. Crivei, Neat and coneat submodules of modules over commutative rings, Bull. Aust. Math. Soc., Abstract .
10. S. Breaz, G. Calugareanu, P. Schultz, Subgroups which admit extensions of homomorphisms, to appear in Forum Math., Abstract, DOI 10.1515/forum-2013-0089
11. S. Breaz, G. Calugareanu, P. Dancev, T. Micu, Nil clean matrix rings, Linear Algebra and Applications, 439, issue 10, 3115-3119 (2013) Abstract
12. C. Valculescu, F. Pop, A note on direct products and Ext1 contravariant functors, Bulletin of the Malaysian Mathematical Science Society, Abstract
13. C. Pelea, I. Purdea, L. Stanca: Fundamental relations in multialgebras. Applications, to appear in European Journal of Combinatorics, DOI: 10.1016/j.ejc.2014.08.015.
14. S. Breaz: A Banach Contraction Principle for Abelian Groups, to appear in Fixed Point Theory.
15. S. Breaz: A Baer-Kaplansky theorem for modules over principal ideal domains, to appear in Journal of Commutative Algebra.
16. G. C. Modoi: The dual of the homotopy category of projective modules satisfies Brown representability, Bulletin of the London Mathematical Society 46 (2014): 765-770 doi:10.1112/blms/bdu039 Abstract
17. S. Breaz, Sigma-pure injectivity and Brown representability, to appear in Proc. Amer. Math. Soc., Abstract
18. C. Pelea, I. Purdea, L. Stanca: Factor multialgebras, universal algebras and fuzzy sets, Carpathian J. Math., accepted (it will be published in vol. 31, 1, 2015)
(B) Papers in international journals or conference proceedings

1. U. Albrecht, S. Breaz, P. Schultz, Functorial properties of Hom and Ext, Contemporary Mathematics, vol. 576, 1-15. Volume

(C) Books

(D) Preprints

1. G. C. Modoi, The dual of Brown representability for some derived categories, preprint. Abstract
2. S. Breaz: When Ext commutes with direct unions, preprint.

Visits, conferences, workshops

2012
1. S. Breaz, Conference on Universal Algebra and Lattice Theory Szeged, Hungary, June 21–25, 2012, talk: Direct products and homomorphisms
2. C. Pelea, Conference on Universal Algebra and Lattice Theory Szeged, Hungary, June 21–25, 2012, talk: Term functions in multialgebra theory
3. C. Valculescu, YOUNG ALGEBRAISTS' CONFERENCE 2012 YAC'12, Groups, Representation Theory and Related Topics, June 11-15, 2012, EPFL, Lausanne, talk: Commuting properties of derived functors.
4. S. Crivei, International Conference on Representations of Algebras (ICRA 2012), Univ. Bielefed, talk: One-sided exact categories.
5. S. Breaz, Workshop on Representations of Algebras (ICRA 2012), Univ. Bielefed.
6. G. C. Modoi, Workshop on Representations of Algebras (ICRA 2012), Univ. Bielefed.
7. C. V\^ alculescu, Conference and Summer School Poisson 2012, Utrecht, July 23 - 27 and July 30 - August 03, 2012.
8. C. Pelea, 84th Workshop on General Algebra, June 8--10, 2012, Dresden; Talk: Multialgebras, factor multialgebras and universal algebras
2013
9. S. Breaz, Joint International Meeting of the AMS and the Romanian Mathematical Society, June 27 - 30, 2013, Alba Iulia, Romania (invited talk at the special session Commutative Algebra: A global Baer-Kaplansky Theorem)
10. S. Breaz, “Rings, categories and Hopf algebras”, dedicated to the 70th birthday of Professor Constantin Nastasescu, University of Bucharest on May 18 and 19, 2013 (talk: On sigma-pure injective modules)
11. S. Crivei, “Rings, categories and Hopf algebras”, dedicated to the 70th birthday of Professor Constantin Nastasescu, University of Bucharest on May 18 and 19, 2013 (talk: One-sided exact categories)
12. S. Crivei, Joint International Meeting of the AMS and the Romanian Mathematical Society, Alba Iulia, Romania, June 27 - 30, 2013 (talk: Finitely accessible categories versus module categories).
13. G. C. Modoi, Joint International Meeting of the AMS and the Romanian Mathematical Society, Alba Iulia, Romania, June 27 - 30, 2013.
14. S. Breaz, Classical Aspects of Ring Theory and Module Theory, Bedlewo, Poland, July 14-20, 2013. (talk: Submodule which admit extensions of homomorphisms)
15. G. C. Modoi, Classical Aspects of Ring Theory and Module Theory, Bedlewo, Poland, July 14-20, 2013. (talk: A module theoretic translation of Brown representabilty for homotopy categories)
16. F. Pop, Joint International Meeting of the AMS and the Romanian Mathematical Society, Alba Iulia, Romania, June 27 - 30, 2013.
17. S. Crivei, Recent Trends in Rings and Algebras. Murcia, June 3-7 (talk: Mitchell Lemma and Gabriel-Popescu Theorem)
18. S. Breaz, Some Trends in Algebra, Prague, September 3-6 (talk: On sigma-pure injective modules)
19. G. C. Modoi, Some Trends in Algebra, Prague, September 3-6 (talk: Some applications of deconstructibility in triangulated categories)
20. C. Pelea: 9th International Conference on Applied Mathematics (ICAM 9), September 25-28, 2013, Baia Mare, Romania (talk: On some particular equivalence relations of multialgebras)
21. S. Breaz: Algebra Seminar, Charles Univ. Prague, October 21, 2013 (talk: Subgroups which admit extensions of homomorphisms)
22. C.G. Modoi: Algebra Seminar, Charles Univ. Prague, October 21, 2013 (talk: On a theorem of G. Bergman).
21. C. Pelea: Algebra Seminar, Charles Univ. Prague, October 21, 2013.
22. F. Pop: Algebra Seminar, Charles Univ. Prague, October 21, 2013.
23. F. Pop, Some Trends in Algebra, Prague, September 3-6, 2013 (talk: On direct products and Ext^1 contravariant functors)

2014

24. S. Breaz: Algebraic Structures and Their Applications, Spineto, June 16-21, 2014 (talk: Commuting properties for the defect functor associated to a homomorphism)
25. S. Breaz: Algebraic Structures and Their Applications, Spineto, June 16-21, 2014 (talk: Salce's lemma in trinagulated categories)
26. F. Pop: Algebraic Structures and Their Applications, Spineto, June 16-21, 2014, participant.
27. S. Breaz: Algebraic Hyperstructures and its Applications, Xanthi, September 2-7, 2014 (talk: Products of Hypergroupoids Associated to Binary Relations)
28. C. Pelea: Algebraic Hyperstructures and its Applications, Xanthi, September 2-7, 2014 (talk: Fundamental Relations in Multialgebras. A Survey)
29. F. Pop: Algebraic Hyperstructures and its Applications, Xanthi, September 2-7, 2014; participant.

Contact

e-mail:
user: bodo server: @math.ubbcluj.ro
name: ssbreaz server: gmail.com
fax:
+40 264 691906
mail:
Simion Breaz
Babes-Bolyai University
Faculty of Mathematics and Computer Science
Str. Mihail Kogalniceanu nr. 1
RO-400084 Cluj-Napoca
Romania