Ring and Module Theory : [Many of these papers were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, during August 18-22, 2008

This volume features a collection of 13 peer-reviewed contributions consisting of expository/survey articles and research papers by 24 authors. Many of theses contributions were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, from August 18 to 22, 2008. The selected contributions examine wide-ranging and cutting-edge developments in various areas of algebra including ring theory, module theory, Hopf algebras, and commutative algebra. The survey articles are written by well-known experts in their respective areas and provide an overview which is useful for researchers in the area as well as those looking for new or related fields to investigate. The research papers give a taste of current research. The variety of topics will be of interest to both graduate students and researchers. Cover......Page 1 Trends in Mathematics......Page 3 Ring and Module Theory......Page 4 ISBN 9783034600064......Page 5 Table of Contents......Page 6 Preface......Page 8 1. Introduction......Page 10 The Classical Hopkins-Levitzki Theorem (H-LT)......Page 11 Extensions of the H-LT......Page 12 2. The Relative H-LT......Page 13 Hereditary torsion theories......Page 14 Relativization......Page 15 Quotient categories and the Gabriel-Popescu Theorem......Page 16 Absolutization......Page 17 A latticial strategy......Page 18 Lattice background......Page 19 The H-LT and Dual H-LT for arbitrary modular lattices......Page 20 The H-LT for upper continuous modular lattices......Page 21 Relative H-LT =⇒ Absolute H-LT......Page 22 The Faith’s counter version of the Relative H-LT......Page 23 Absolute H-LT ⇐⇒ Classical H-LT......Page 24 6. The Absolute and Relative Dual H-LT......Page 25 The definition of the Krull dimension of a poset......Page 26 The definition of the dual Krull dimension of a poset......Page 27 A Krull dimension-like extension of the Absolute H-LT......Page 28 Localization of modular lattices......Page 29 Sketch of the proof of Theorem 7.3......Page 31 8. Four open problems......Page 32 References......Page 33 1. Introduction......Page 36 2. Brief history of hulls......Page 40 3. Definitions of a ring hull......Page 43 4. Existence and uniqueness of ring hulls......Page 48 5. Transference between R and overrings......Page 61 6. How does Q(R) determine R?......Page 64 7. Hulls of ring extensions......Page 66 8. Modules with FI-extending hulls......Page 68 9. Applications to rings with involution......Page 71 References......Page 76 1. Introduction and preliminaries......Page 82 2. Proof of Theorem 1.1......Page 84 3. Grobner-Shirshov basis of An......Page 86 References......Page 89 1. Introduction......Page 92 3. The isomorphism problem......Page 93 3.1. Preliminary results......Page 94 4. The main result......Page 96 4.1. A special case......Page 97 References......Page 99 Introduction......Page 100 1. Frobenius and symmetric algebras......Page 104 2. Character algebras of semisimple Hopf algebras......Page 108 3. C(H) is a commutative algebra......Page 111 4. Factorizable Hopf algebras......Page 118 References......Page 122 1. Introduction......Page 124 2. Equivalent conditions to r.w.r.......Page 125 3. Examples and constructions......Page 127 4. Related conditions......Page 129 References......Page 131 2. Regular homomorphisms and the total......Page 134 4. The radicals for HomR(A,M)......Page 135 5. Regular substructures of Hom......Page 136 References......Page 137 1. Definitions, notations and preliminaries......Page 138 2. Main results......Page 140 References......Page 143 1. Introduction......Page 144 2. FP-injective complexes......Page 145 References......Page 150 1. Introduction......Page 152 2. T-projective dimension......Page 153 3. Tnm-injective modules and Tnm-projective modules......Page 155 References......Page 157 1. Subrings of Artinian rings......Page 158 2. Additive rank functions......Page 161 3. Chain conditions......Page 165 4. Modules with the direct sum condition......Page 168 References......Page 171 1. Preliminaries and summary of results......Page 174 2. (α, β)-higher derivations......Page 176 3. Higher differentiation invariance......Page 177 4. Extending derivation to different modules of quotients......Page 179 5. Symmetric modules of quotients......Page 180 6. Torsion theory that is not differential......Page 182 References......Page 183 0. Introduction......Page 184 1. Bad behavior......Page 186 2. Affine domains of dimension two......Page 188 3. Polynomial rings over semilocal one-dimensional domains......Page 194 4. Two-dimensional power series rings......Page 196 References......Page 200 Participants......Page 204 A Seventy Years Jubilee: The Hopkins-levitzki Theorem / T. Albu -- A Theory Of Hulls For Rings And Modules / G.f.birkenmeier, J.k.park And S. Tariq Rizvi -- Rewriting As A Special Case Of Noncommutative Gröbner Bases Theory For The Affine Weyl Group Ãn / A.s. Çevik, C. Özel And E.g. Karpuz -- A Classification Of A Certain Class Of Completely Primary Finite Rings / C.j. Chikunji -- Higman Ideals And Verlinde-type Formulas For Hopf Algebras / M. Cohen And S. Westreich -- Right Weakly Regular Rings : A Survey / H.e. Heatherly And R.p.tucci -- Substructures Of Hom / F. Kasch -- Weak Lifting Modules With Small Radical / D. Keskin Tütüncü And S.h. Mohamed -- F P-injective Complexes / L. Mao And N. Ding -- A Generalization Of Homological Dimensions / M.j. Nikmehr And F. Shaveisi -- Additive Rank Functions And Chain Conditions / P.f. Smith -- A Note On ([alpha, Beta] )-higher Derivations And Their Extensions To Modules Of Quotients / L. Vas̆ And C. Papachristou -- Prime Ideals In Noetherian Rings: A Survey / R. Wiegand And S. Wiegand. Toma Albu ... [et Al.], Editors. Selected Conference Papers. Includes Bibliographical References. The selected contributions examine wide-ranging and cutting-edge developments in various areas of algebra including ring theory, module theory, Hopf algebras, and commutative algebra. The survey articles are written by well-known experts in their respective areas and provide an overview which is useful for researchers in the area as well as those looking for new or related fields to investigate. The research papers give a taste of current research. The variety of topics will be of interest to both graduate students and researchers. --Book Jacket The volume consists of a collection of invited research papers and expository/survey articles, many of which were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, in August 2008. The selected papers and articles examine wide ranging and cutting edge developments in various areas of algebra, including ring theory, module theory, and commutative algebra. There is also a section devoted to open problems to motivate further research.