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Generalities on algebraic structures (from magmas to finitely presented associative rings).
Examples of important rings and algebras.
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Grading on an algebraic structure. Properties and invariants of graded rings and their ideals.
Algorithmic aspects and noncommutative factorization.
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Gröbner bases of ideals in free associative algebra.
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Filtration on an algebraic structure and the associated grading. Filtered-graded transfer of
ring-theoretic properties (like theorems of Jacobson). Finite dimensional and good
filtrations. Hilbert-Poincare series of rings and modules.
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Dimension functions: definition, properties (incl. exactness), formalism.
Examples and non-examples of dimensions, their exactness. The holonomic number
of a module category. Equidimensional and co-quidimensional modules.
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Krull dimension over commutative rings, its properties and the connection to Hilbert series.
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Gel'fand-Kirillov dimension (GKdim) over K-algebras. Its properties and behaviour on subalgebras,
submodules and its extension beyond finitely presented case. GKdim on commutative algebras and
comparison with the Krull dimension.
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Limited exactness of GKdim and examples of categories where exactess is proven. Computation of GKdim: several approaches. The connection of GKdim to Hilbert series; more on GKdim-holonomic modules and Bernstein-Hilbert multiplicity.
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