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Grant CEEX ET-47, April 2006-March 2008
"Generalizations of injectivity and approximations in abelian categories"  

Financed by the National Authority for Scientific Research
Faculty of Mathematics and Computer Science,  "Babes-Bolyai" University, Cluj-Napoca
Research team
Assoc. Prof. Dr. Septimiu Crivei (project manager), Assoc. Prof. Dr. Simion Breaz, Assist. Prof. Dr. Ciprian Modoi, Assist. Prof. Dr. Gabriela Olteanu, Researcher Dan Salajan.

Project summary and objectives

The project deals with several research directions in the nowadays algebra: the study of some special classes of objects in abelian categories, theory of approximation through various envelopes and covers, with consequences in tilting and cotilting theory, torsion theories in abelian categories, theory of abelian groups. The project aims to revaluate and develop the research skills of the involved young researchers and to offer a competitive information and training framework for the involved doctoral students. Also, the project has as finality the substantial improvement of the available infrastructure, in view of providing a modern logistic (both computers and scientific literature) needed for research.

Scientific objectives of the project:

2006

1. Establishing properties of some generalizations of the classes Add(M) and Prod(M).
2. Definition and study of some generalizations of extending modules.
3. Establishing characterizations and properties of purely extending and self-c-injective modules.
4. Study of exact functors modulo Serre classes and torsion theories.

2007

1. Dualizations.
2. Versions for torsion theories.
3. Determination and characterization of some functors with exactness properties modulo torsion theories.
4. Study of some dual notions for torsion theories.
5. Determination and characterization of some functors with exactness properties modulo Serre classes.
6. Study of some special properties in triangulated categories.

2008

1. Establishing some special properties in triangulated categories.
2. Concluding the proposed research. 

Results
11 published papers (10 papers in ISI journals), 1 homologated computer package, 1 communicated computer package.
Their abstracts are here.

Papers in ISI journals
1. I. Crivei, S. Crivei, I. Purdea, Change of ring and torsion-theoretic injectivity, Bull. Austral. Math. Soc. 75 (2007), 127-133. 
2. S. Caenepeel, S. Crivei, A. Marcus, M. Takeuchi, Morita equivalences induced by bimodules over Hopf-Galois extensions, J. Algebra 314 (2007), 267-302.  doi:10.1016/j.jalgebra.2007.02.033 
3. S. Breaz, J. Zemlicka, When every self-small module is finitely generated, J. Algebra 315 (2007), 885-893. doi:10.1016/j.jalgebra.2007.01.037 
4. U. Albrecht, S. Breaz, W. Wickless, Purity and self-small groups, Comm. Algebra 35 (2007), 3789-3807. doi:10.1080/00927870701509545
5. G. Olteanu, A. del Rio, Group algebras of Kleinian type and groups of units, J. Algebra 318 (2007), 856-870. doi:10.1016/j.jalgebra.2007.03.026 
6. S. Crivei, Sigma-extending modules, Sigma-lifting modules, and proper classes, Comm. Algebra 36 (2008), 529-545. doi:10.1080/00927870701718930
7. G. Olteanu, A. del Rio, An algorithm to compute the Wedderburn decomposition of semisimple group algebras implemented in the GAP package wedderga, J. Symbolic Comput. 44 (2009), 507-516. doi:10.1016/j.jsc.2007.07.019
8. A. Herman, G. Olteanu, A. del Rio, Ring isomorphism of cyclic cyclotomic algebras, Algebr. Represent. Theory 12 (2009), 365-370. doi:10.1007/s10468-009-9158-1.
9. S. Crivei, Relatively extending modules, Algebr. Represent. Theory 12 (2009), 319-332. doi:10.1007/s10468-009-9155-4
10. G. Modoi, Generalized lax epimorphisms in the additive case, J. Pure Appl. Algebra 215 (2011), 697-704. doi:10.1016/j.jpaa.2010.06.018

Other papers

11. S. Breaz, The number of Remak decompositions of a finite Abelian group, Mathematica (Cluj) 50(73) (2008), 159-168.

Computer packages

12. O. Broche Cristo, A. Konovalov, A. Olivieri, G. Olteanu, A. del Rio, Wedderga - Wedderburn Decomposition of Group Algebras, Version 4.3; 2007 (http://www.gap-system.org/Packages/wedderga.html, http://www.um.es/adelrio/wedderga.htm). The package was homologated and included in the official distribution of the GAP system.
13. S. Crivei, G. Olteanu, S. Suteu-Szollosi, ELISA - A collection of GAP algorithms related to extending and lifting abelian groups, Version 1.1; 2007 (http://www.gap-system.org/Packages/undep.html, http://math.ubbcluj.ro/~crivei/GAP_project). The package was communicated to the GAP group.

Talks given at conferences and research seminars

1. Conference on Lattice Theory, Budapest, Hungary, June 6-9, 2006; S. Breaz, talk with the title: Every abelian group is determined by a subgroup lattice.
2. International Congress of Mathematicians, Madrid, Spain, August 22-30, 2006; S. Crivei, poster with the title: Extending modules versus purely extending modules.
3. Non-commutative Algebra, Granada, Spain, August 31 - September 6, 2006; S. Crivei, talk with the title: Extending modules versus purely extending modules.
4. Non-commutative Algebra, Granada, Spain, August 31 - September 6, 2006; G. Olteanu, talk with the title: Computing the Wedderburn decomposition of group algebras by the Brauer-Witt Theorem.
5. Tenth Meeting on Computer Algebra and Applications, Sevilla, Spain, September 7-9, 2006; G. Olteanu, coauthor of the talk with the title: Wedderga: Wedderburn decomposition of group algebras (A. del Rio).
6. New Techniques in Hopf Algebras and Graded Ring Theory, Brusells, Belgium, September 19-23, 2006; S. Crivei, coauthor of the invited talk with the title: Morita equivalences induced by bimodules over Hopf-Galois extensions (A. Marcus).
7. Algebra Seminar - Vrije Universiteit, Brusells, Belgium, May 25, 2007; G. Olteanu, invited talk with the title: Ring isomorphism of cyclic cyclotomic algebras.
8. Abelian Groups And Modules Over Commutative Rings, Connecticut, USA, June 11-15, 2007; C. Modoi, invited talk with the title: When a functor does induce an abelian localization?
9. Algebraic and Combinatorial Methods in Concrete Classes of Algebras and Groups, Alden Biesen, Belgium, September 2-7, 2007; G. Olteanu, coauthor of the invited talk: The subgroup of the Schur group generated by cyclic cyclotomic algebras (A. del Rio).
10. Some Trends in Algebra '07, Prague, Czech Republic, September 3-7, 2007; S. Breaz, talk with the title: Warfield duality for mixed groups.
11. International Conference on Noncommutative Rings and Geometry, Almeria, Spain, September 18-22, 2007; S. Crivei, talk with the title: Extending and lifting type modules.
12. Algebra Seminar- University of Granada, Granada, Spain, October 10, 2007; S. Crivei, invited talk with the title: Covers and envelopes by Ext-orthogonal classes.
13. Mini-workshop: Arithmetik von Gruppenringen, Oberwolfach, Germany, November 25 - December 1, 2007; G. Olteanu, invited talk with the title: Wedderburn decomposition of group algebras and Schur groups.
14. Categorical Methods for Rings and Modules, Murcia, Spain, December 3-6, 2007; G. Olteanu, talk with the title: The subgroup of the Schur group generated by cyclic cyclotomic algebras.

Other participations in conferences
15. Conference on Lattice Theory, Budapest, Hungary, June 6-9, 2006; D. Salajan.
16. International Congress of Mathematicians, Madrid, Spain, August 22-30, 2006; G. Olteanu.
17. International Conference on Modules and Comodules, Porto, Portugal, September 8-10, 2006; S. Crivei.
18. International Conference on Noncommutative Rings and Geometry, Almeria, Spain, September 18-22, 2007; G. Olteanu.
19. Categorical Methods for Rings and Modules, Murcia, Spain, December 3-6, 2007; S. Crivei.

Contact data
Address: Septimiu Crivei, Faculty of Mathematics and Computer Science, Babes-Bolyai University, Str. M. Kogalniceanu 1, 400084 Cluj-Napoca, Romania
E-mail: crivei at math dot ubbcluj dot ro,  scrivei at gmail dot com
Tel.: ++40-264-405327
Fax: ++40-264-591906