Monograph 01, 2015.

Chain conditions in modular lattices with applications to Grothendieck categories and torsion theories

Toma Albu

Professor Toma Albu is a Senior researcher at the Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania. His research interests involve Ring Theory, Module Theory, Field Theory, Algebraic Theory of Lattices, and Algebraic Number Theory. He received the M.Sc. (1966) and Ph.D. (1971) degrees in Mathematics from the Bucharest University. Dr. Albu has authored or coauthored several books, including "Relative Finiteness in Module Theory" (1984) and "Cogalois Theory" (2003), Marcel Dekker, and over 100 papers appearing in various international journals. He was a Humboldt Research Fellow at the Universities of Munich and Dusseldorf. Dr. Albu has also held Visiting Professor positions at the Osaka City University (Japan), Padua University (Italy), University of Wisconsin, Milwaukee (USA), The Ohio State University, Columbus (USA), and University of California, Santa Barbara (USA). He is Vice- President of the Romanian Mathematical Society and member of the American Mathematical Society. The text presents in a compact way some basics of Lattice Theory with a great emphasis on chain conditions in modular lattices, that are then applied to Grothendieck categories and module categories equipped with hereditary torsion theories to obtain immediately and in a unified manner significant results in these areas. It also includes other results of Algebraic Theory of Lattices that are interesting in their own right. The renowned Hopkins-Levitzki Theorem and Osofsky-Smith Theorem from Ring and Module Theory are among the most relevant illustrations of a general strategy which consists on putting a module-theoretical concept/result into a latticial frame (latticization), in order to translate that concept/result to Grothendieck categories (absolutization) and module categories equipped with hereditary torsion theories (relativization).
	2010 Mathematics Subject Classification: Primary: 06-01, 06-02, 06C05, 16-01, 16-02, 16D90; Secondary: 16S90, 18E15, 18E40.

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Toma Albu Simion Stoilow Institute of Mathematics of the Romanian Academy Research Unit 5 P.O. Box 1 - 764 RO - 010145 Bucharest 1, ROMANIA E-mail address: Toma.Albu@imar.ro

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