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Commutative algebra. --- Associative algebras. --- Separable algebras. --- Algebras, Separable --- Associative algebras --- Algebras, Associative --- Algebra

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Commutative algebra. --- Separable algebras. --- Algebras, Separable --- Associative algebras --- Algebra

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Tilting theory originates in the representation theory of finite dimensional algebras. Today the subject is of much interest in various areas of mathematics, such as finite and algebraic group theory, commutative and non-commutative algebraic geometry, and algebraic topology. The aim of this book is to present the basic concepts of tilting theory as well as the variety of applications. It contains a collection of key articles, which together form a handbook of the subject, and provide both an introduction and reference for newcomers and experts alike.

Associative algebras. --- Modules (Algebra) --- Representations of algebras. --- Dimension theory (Algebra) --- Finite groups. --- Groups, Finite --- Group theory --- Associative algebras --- Commutative algebra --- Algebra --- Finite number systems --- Modular systems (Algebra) --- Finite groups --- Rings (Algebra) --- Algebras, Associative

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Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.

Rings (Algebra) --- Ideals (Algebra) --- Algebraic ideals --- Algebraic fields --- Algebraic rings --- Ring theory --- Associative algebras. --- Algebras, Associative --- Algebra

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The relations between Frobenius manifolds and singularity theory are treated here in a rigorous yet accessible manner. For those working in singularity theory or other areas of complex geometry, this book will open the door to the study of Frobenius manifolds. This class of manifolds are now known to be relevant for the study of singularity theory, quantum cohomology, mirror symmetry, symplectic geometry and integrable systems. The first part of the book explains the theory of manifolds with a multiplication on the tangent bundle. The second presents a simplified explanation of the role of Frobenius manifolds in singularity theory along with all the necessary tools and several applications. Readers will find here a careful and sound study of the fundamental structures and results in this exciting branch of maths. This book will serve as an excellent resource for researchers and graduate students who wish to work in this area.

Singularities (Mathematics) --- Frobenius algebras. --- Moduli theory. --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebras, Frobenius --- Associative algebras