An introduction to rings and modules with K-theory in view

Contents......Page all_28847_to_00281.cpc0007.djvu
Preface......Page all_28847_to_00281.cpc0013.djvu
1.1 Rings......Page all_28847_to_00281.cpc0017.djvu
1.1.1 The definition......Page all_28847_to_00281.cpc0018.djvu
1.1.2 Nonunital rings......Page all_28847_to_00281.cpc0019.djvu
1.1.4 Ideals......Page all_28847_to_00281.cpc0020.djvu
1.1.6 Homomorphisms......Page all_28847_to_00281.cpc0021.djvu
1.1.8 Residue rings......Page all_28847_to_00281.cpc0023.djvu
1.1.11 Units......Page all_28847_to_00281.cpc0024.djvu
1.1.12 Constructing the field of fractions......Page all_28847_to_00281.cpc0025.djvu
1.1.13 Noncommutative polynomials......Page all_28847_to_00281.cpc0026.djvu
Exercises......Page all_28847_to_00281.cpc0027.djvu
1.2 Modules......Page all_28847_to_00281.cpc0030.djvu
1.2.1 The definition......Page all_28847_to_00281.cpc0031.djvu
1.2.2 Some first examples......Page all_28847_to_00281.cpc0032.djvu
1.2.4 Homomorphisms of modules......Page all_28847_to_00281.cpc0033.djvu
1.2.5 The composition of homomorphisms......Page all_28847_to_00281.cpc0034.djvu
1.2.6 The opposite of a ring......Page all_28847_to_00281.cpc0035.djvu
1.2.7 Balanced bimodules......Page all_28847_to_00281.cpc0036.djvu
1.2.8 Submodules and generators......Page all_28847_to_00281.cpc0037.djvu
1.2.10 Quotient modules......Page all_28847_to_00281.cpc0038.djvu
1.2.12 Images and inverse images......Page all_28847_to_00281.cpc0039.djvu
1.2.16 Irreducible modules......Page all_28847_to_00281.cpc0040.djvu
1.2.18 Maximal elements in ordered sets......Page all_28847_to_00281.cpc0041.djvu
1.2.23 Torsion-free modules and spaces over the field of fractions......Page all_28847_to_00281.cpc0043.djvu
Exercises......Page all_28847_to_00281.cpc0044.djvu
2 Direct Sums and Short Exact Sequences......Page all_28847_to_00281.cpc0052.djvu
2.1.1 Internal direct sums......Page all_28847_to_00281.cpc0053.djvu
2.1.2 Examples: vector spaces......Page all_28847_to_00281.cpc0054.djvu
2.1.3 Examples: abelian groups......Page all_28847_to_00281.cpc0055.djvu
2.1.5 External direct sums......Page all_28847_to_00281.cpc0056.djvu
2.1.6 Standard inclusions and projections......Page all_28847_to_00281.cpc0057.djvu
2.1.9 Idempotents......Page all_28847_to_00281.cpc0058.djvu
2.1.11 Infinite direct sums......Page all_28847_to_00281.cpc0059.djvu
2.1.13 Remarks......Page all_28847_to_00281.cpc0060.djvu
2.1.14 Ordered index sets......Page all_28847_to_00281.cpc0061.djvu
2.1.15 The module L^A......Page all_28847_to_00281.cpc0062.djvu
2.1.17 Left-handed notation......Page all_28847_to_00281.cpc0063.djvu
2.1.19 Free modules......Page all_28847_to_00281.cpc0064.djvu
2.1.21 Extending maps......Page all_28847_to_00281.cpc0065.djvu
Exercises......Page all_28847_to_00281.cpc0067.djvu
2.2 Matrices, Bases, Homomorpihsms of Free Modules......Page all_28847_to_00281.cpc0070.djvu
2.2.1 Bases......Page all_28847_to_00281.cpc0071.djvu
2.2.3 Coordinates......Page all_28847_to_00281.cpc0072.djvu
2.2.5 Matrices for homomorphisms......Page all_28847_to_00281.cpc0074.djvu
2.2.7 Change of basis ,......Page all_28847_to_00281.cpc0075.djvu
2.2.9 Matrices of endomorphisms......Page all_28847_to_00281.cpc0076.djvu
2.2.10 Normal forms of matrices......Page all_28847_to_00281.cpc0077.djvu
2.2.12 Scalar matrices and endomorphisms......Page all_28847_to_00281.cpc0078.djvu
2.2.13 Infinite bases......Page all_28847_to_00281.cpc0080.djvu
2.2.14 Free left modules......Page all_28847_to_00281.cpc0081.djvu
Exercises......Page all_28847_to_00281.cpc0082.djvu
2.3 Invariant Basis Number......Page all_28847_to_00281.cpc0085.djvu
2.3.3 Two non-square invertible matrices......Page all_28847_to_00281.cpc0086.djvu
2.3.4 The type......Page all_28847_to_00281.cpc0087.djvu
Exercises......Page all_28847_to_00281.cpc0088.djvu
2.4.1 The definition......Page all_28847_to_00281.cpc0091.djvu
2.4.3 Short exact sequences......Page all_28847_to_00281.cpc0092.djvu
2.4.4 Direct sums and splittings......Page all_28847_to_00281.cpc0093.djvu
2.4.6 Dual numbers......Page all_28847_to_00281.cpc0095.djvu
2.4.8 Pull-backs and push-outs......Page all_28847_to_00281.cpc0096.djvu
2.4.10 Base change for short exact sequences......Page all_28847_to_00281.cpc0098.djvu
2.4.11 The direct sum of short exact sequences......Page all_28847_to_00281.cpc0099.djvu
Exercises......Page all_28847_to_00281.cpc0100.djvu
2.5.1 The definition and basic properties......Page all_28847_to_00281.cpc0105.djvu
2.5.9 Idempotents and projective modules......Page all_28847_to_00281.cpc0109.djvu
2.5.13 Injective modules......Page all_28847_to_00281.cpc0110.djvu
Exercises......Page all_28847_to_00281.cpc0111.djvu
2.6 Direct Products of Rings......Page all_28847_to_00281.cpc0113.djvu
2.6.1 The definition......Page all_28847_to_00281.cpc0114.djvu
2.6.2 Central idempotents......Page all_28847_to_00281.cpc0115.djvu
2.6.4 Remarks......Page all_28847_to_00281.cpc0116.djvu
2.6.6 Modules......Page all_28847_to_00281.cpc0117.djvu
2.6.7 Homomorphisms......Page all_28847_to_00281.cpc0118.djvu
2.6.10 Historical note......Page all_28847_to_00281.cpc0120.djvu
Exercises......Page all_28847_to_00281.cpc0121.djvu
3.1 Noetherian Rings......Page all_28847_to_00281.cpc0124.djvu
3.1.1 The Noetherian condition......Page all_28847_to_00281.cpc0125.djvu
3.1.5 The ascending chain condition and the maximum condition......Page all_28847_to_00281.cpc0126.djvu
3.1.11 Module-finite extensions......Page all_28847_to_00281.cpc0128.djvu
Exercises......Page all_28847_to_00281.cpc0130.djvu
3.2 Skew Polynomial Rings......Page all_28847_to_00281.cpc0131.djvu
3.2.1 The definition......Page all_28847_to_00281.cpc0132.djvu
3.2.2 Some endomorphisms......Page all_28847_to_00281.cpc0134.djvu
3.2.7 Euclidean domains......Page all_28847_to_00281.cpc0136.djvu
3.2.11 Euclid\'s algorithm......Page all_28847_to_00281.cpc0138.djvu
3.2.12 An example......Page all_28847_to_00281.cpc0139.djvu
3.2.13 Inner order and the centre......Page all_28847_to_00281.cpc0140.djvu
3.2.15 Ideals......Page all_28847_to_00281.cpc0141.djvu
3.2.19 Total division......Page all_28847_to_00281.cpc0142.djvu
3.2.22 Unique factorization......Page all_28847_to_00281.cpc0143.djvu
3.2.23 Further developments......Page all_28847_to_00281.cpc0144.djvu
Exercises......Page all_28847_to_00281.cpc0145.djvu
3.3 Modules over Skew Polynomial Rings......Page all_28847_to_00281.cpc0148.djvu
3.3.1 Elementary operations......Page all_28847_to_00281.cpc0149.djvu
3.3.3 Rank and invariant factors......Page all_28847_to_00281.cpc0150.djvu
3.3.4 The structure of modules......Page all_28847_to_00281.cpc0151.djvu
3.3.7 Rank and invariant factors for modules......Page all_28847_to_00281.cpc0153.djvu
3.3.8 Non-cancellation......Page all_28847_to_00281.cpc0154.djvu
Exercises......Page all_28847_to_00281.cpc0156.djvu
4.1 Artinian Modules......Page all_28847_to_00281.cpc0161.djvu
4.1.2 Examples......Page all_28847_to_00281.cpc0162.djvu
4.1.3 Fundamental properties......Page all_28847_to_00281.cpc0163.djvu
4.1.8 Composition series......Page all_28847_to_00281.cpc0164.djvu
4.1.11 Multiplicity......Page all_28847_to_00281.cpc0167.djvu
4.1.13 Reducibility......Page all_28847_to_00281.cpc0168.djvu
4.1.15 Complete reducibility......Page all_28847_to_00281.cpc0170.djvu
4.1.22 Fully invariant submodules......Page all_28847_to_00281.cpc0173.djvu
4.1.24 The socle series......Page all_28847_to_00281.cpc0174.djvu
Exercises......Page all_28847_to_00281.cpc0175.djvu
4.2 Artinian Semisimple Rings......Page all_28847_to_00281.cpc0177.djvu
4.2.1 Definitions and the statement of the Wedderbum-Artin Theorem......Page all_28847_to_00281.cpc0178.djvu
4.2.5 Matrix rings......Page all_28847_to_00281.cpc0179.djvu
4.2.6 Products of matrix rings......Page all_28847_to_00281.cpc0180.djvu
4.2.11 Finishing the proof of the Wedderburn-Artin Theorem......Page all_28847_to_00281.cpc0182.djvu
4.2.16 Recapitulation of the argument......Page all_28847_to_00281.cpc0185.djvu
Exercises......Page all_28847_to_00281.cpc0186.djvu
4.3.1 The Jacobson radical......Page all_28847_to_00281.cpc0189.djvu
4.3.3 Basic properties......Page all_28847_to_00281.cpc0190.djvu
4.3.11 Alternative descriptions of the Jacobson radical......Page all_28847_to_00281.cpc0192.djvu
4.3.19 Nilpotent ideals and a characterization of Artinian rings......Page all_28847_to_00281.cpc0195.djvu
4.3.22 Semilocal rings......Page all_28847_to_00281.cpc0196.djvu
4.3.24 Local rings......Page all_28847_to_00281.cpc0197.djvu
Exercises......Page all_28847_to_00281.cpc0198.djvu
5 Dedekind Domains......Page all_28847_to_00281.cpc0202.djvu
5.1.1 Prime ideals......Page all_28847_to_00281.cpc0203.djvu
5.1.4 Coprime ideals......Page all_28847_to_00281.cpc0204.djvu
5.1.7 Fractional ideals......Page all_28847_to_00281.cpc0205.djvu
5.1.10 Dedekind domains - the definition......Page all_28847_to_00281.cpc0206.djvu
5.1.11 The class group......Page all_28847_to_00281.cpc0207.djvu
5.1.13 An exact sequence......Page all_28847_to_00281.cpc0208.djvu
5.1.15 Ideal theory in a Dedekind domain......Page all_28847_to_00281.cpc0209.djvu
5.1.25 Principal ideal domains......Page all_28847_to_00281.cpc0212.djvu
Exercises......Page all_28847_to_00281.cpc0213.djvu
5.2.1 Integers......Page all_28847_to_00281.cpc0216.djvu
5.2.6 Quadratic fields......Page all_28847_to_00281.cpc0219.djvu
5.2.9 Separability and integral closure......Page all_28847_to_00281.cpc0221.djvu
Exercises......Page all_28847_to_00281.cpc0224.djvu
5.3.1 Factorization in general......Page all_28847_to_00281.cpc0226.djvu
5.3.2 Factorization in the quadratic case......Page all_28847_to_00281.cpc0227.djvu
5.3.3 Explicit factorizations......Page all_28847_to_00281.cpc0229.djvu
5.3.4 A summary......Page all_28847_to_00281.cpc0230.djvu
5.3.5 The norm......Page all_28847_to_00281.cpc0231.djvu
5.3.9 The Euclidean property......Page all_28847_to_00281.cpc0232.djvu
5.3.10 The class group again......Page all_28847_to_00281.cpc0233.djvu
5.3.15 Computations of class groups......Page all_28847_to_00281.cpc0234.djvu
5.3.21 Function fields......Page all_28847_to_00281.cpc0237.djvu
Exercises......Page all_28847_to_00281.cpc0238.djvu
6 Modules over Dedekind Domains......Page all_28847_to_00281.cpc0240.djvu
6.1 Projective Modules over Dedekind Domains......Page all_28847_to_00281.cpc0241.djvu
6.1.7 The standard form......Page all_28847_to_00281.cpc0244.djvu
Exercises......Page all_28847_to_00281.cpc0245.djvu
6.2 Valuation Rings......Page all_28847_to_00281.cpc0246.djvu
6.2.1 Valuations......Page all_28847_to_00281.cpc0247.djvu
6.2.4 Localization......Page all_28847_to_00281.cpc0248.djvu
6.2.10 The localization as a Euclidean domain......Page all_28847_to_00281.cpc0250.djvu
Exercises......Page all_28847_to_00281.cpc0251.djvu
6.3 Torsion Modules over Dedekind Domains......Page all_28847_to_00281.cpc0253.djvu
6.3.1 Torsion modules......Page all_28847_to_00281.cpc0254.djvu
6.3.7 Primary modules......Page all_28847_to_00281.cpc0255.djvu
6.3.10 Elementary divisors......Page all_28847_to_00281.cpc0257.djvu
6.3.13 Primary decomposition......Page all_28847_to_00281.cpc0259.djvu
6.3.17 Elementary divisors again......Page all_28847_to_00281.cpc0260.djvu
6.3.18 Homomorphisms......Page all_28847_to_00281.cpc0261.djvu
6.3.21 Alternative decompositions......Page all_28847_to_00281.cpc0262.djvu
6.3.22 Homomorphisms again......Page all_28847_to_00281.cpc0263.djvu
Exercises......Page all_28847_to_00281.cpc0265.djvu
References......Page all_28847_to_00281.cpc0268.djvu
Index......Page all_28847_to_00281.cpc0273.djvu