THE PRINCIPLES OF ELEMENTARY ALGEBRA

By AMELIA GONZALEZ

Chapter 1 Elementary properties of rings 1 Basic notions 1 2 Special elements and ideals in rings 7 3 Special rings 14 4 Chain conditions for rings 26 5 Algebras and group rings 30 Chapter 2 Module categories 6 Elementary properties of modules 36 7 The category of R-modules 43 8 Internal direct sum 57 9 Product, coproduct and subdirect product 64 10 Pullback and pushout 73 11 Functors, Hom-functors 80 12 Tensor product, tensor functor 90 Chapter 3 Modules characterized by the Hom-functor 13 Generators, trace 105 14 Cogenerators, reject 112 15 Subgenerators, the category ] 118 16 Injective modules 127 17 Essential extensions, injective hulls 137 18 Projective modules 148 19 Superfluous epimorphisms, projective covers 159 1 36 105 Chapter 4 Notions derived from simple modules 20 Semisimple modules and rings 165 21 Socle and radical of modules and rings 174 22 The radical of endomorphism rings 185 23 Co-semisimple and good modules and rings 190 Chapter 5 Finiteness conditions in modules 24 The direct limit 196 25 Finitely presented modules 207 26 Coherent modules and rings 214 27 Noetherian modules and rings 221 28 Annihilator conditions 230 Chapter 6 Dual finiteness conditions 29 The inverse limit 238 30 Finitely copresented modules 248 31 Artinian and co-noetherian modules 253 32 Modules of finite length 265 Chapter 7 Pure sequences and derived notions 33 P-pure sequences, pure projective modules 274 34 Purity in ], R-MOD and ZZ-MOD 281 35 Absolutely pure modules 297 36 Flat modules 304 37 Regular modules and rings 313 38 Copure sequences and derived notions 322 Chapter 8 Modules described by means of projectivity 39 (Semi)hereditary modules and rings 328 40 Semihereditary and hereditary domains 341 41 Supplemented modules 348 42 Semiperfect modules and rings 371 43 Perfect modules and rings 382 Chapter 9 Relations between functors 44 Functorial morphisms 393 45 Adjoint pairs of functors 399 46 Equivalences of categories 413 47 Dualities between categories 425 48 Quasi-Frobenius modules and rings 445 165 196 238 274 328 393 Chapter 10 Functor rings 49 Rings with local units 464 50 Global dimensions of modules and rings 476 51 The functor bHom(V,?) 485 52 Functor rings of ] and R-MOD 506 53 Pure semisimple modules and rings 521 54 Modules of finite representation type 531 55 Serial modules and rings 539 56 Homo-serial modules and rings 560