Items related to Algorithmic Methods in Non-Commutative Algebra: Applications...
Synopsis
Introduction. 1: Generalities on rings. 1. Rings and ideals. 2. Modules and chain conditions. 3. Ore extensions. 4. Factorization. 5. Other examples. 6. Quantum groups. 2: Gröbner basis computation algorithms. 1. Admissible orders. 2. Left Poincaré-Birkhoff-Witt Rings. 3. Examples. 4. The Division Algorithm. 5. Gröbner bases for left ideals. 6. Buchberger's Algorithm. 7. Reduced Gröbner Bases. 8. Poincaré-Birkhoff-Witt rings. 9. Effective computations for two-sided ideals. 3: Poincaré-Birkhoff-Witt Algebras. 1. Bounding quantum relations. 2. Misordering. 3. The Diamond Lemma. 4. Poincaré-Birkhoff-Witt Theorems. 5. Examples. 6. Iterated Ore Extensions. 4: First applications. 1. Applications to left ideals. 2. Cyclic finite-dimensional modules. 3. Elimination. 4. Graded and filtered algebras. 5. The omega-filtration of a PBW algebra. 6. Homogeneous Gröbner bases. 7. Homogenization. 5: Gröbner bases for modules. 1. Gröbner bases and syzygies. 2. Computation of the syzygy module. 3. Admissible orders in stable subsets. 4. Gröbner bases for modules. 5. Gröbner bases for subbimodules. 6. Elementary applications of Gröbner bases for modules. 7. Graded and filtered modules. 8. The omega-filtration of a module. 9. Homogeneous Gröbner bases. 10. Homogenization. 6:Syzygies and applications. 1. Syzygies for modules. 2. Intersections. 3. Applications to finitely presented modules. 4. Schreyer's order. 5. Free resolutions. 6. Computation of Hom and Ext. 7: The Gelfand-Kirillov dimension and the Hilbert polynomial. 1. The Gelfand-Kirillov dimension. 2. The Hilbert function of a stable subset. 3.The Hilbert function of a module over a PBW algebra. 4. The Gelfand-Kirillov dimension of PBW algebras. 8: Primality. 1. Localization. 2. The Ore condition and syzygies. 3. A primality test. 4. The primality test in iterated differential operator rings. 5. The primality test in coordinate rings of quantum spaces. Index. References.
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