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A ring is called quasi-Frobenius if it is right or left selfinjective, and right or left artinian (all four combinations are equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field, and the subject is intimately related to duality, the duality from right to left modules induced by the hom functor and the duality related to annihilators. The present extent of the theory is vast, and this book makes no attempt to be encyclopedic; instead it provides an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field.

Modules (Algebra). --- Quasi-Frobenius rings. --- Ring extensions (Algebra). --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- QF rings --- Rings, QF --- Rings, Quasi-Frobenius --- Associative rings

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Integral Closure gives an account of theoretical and algorithmic developments on the integral closure of algebraic structures. These are shared concerns in commutative algebra, algebraic geometry, number theory and the computational aspects of these fields. The overall goal is to determine and analyze the equations of the assemblages of the set of solutions that arise under various processes and algorithms. It gives a comprehensive treatment of Rees algebras and multiplicity theory - while pointing to applications in many other problem areas. Its main goal is to provide complexity estimates by tracking numerically invariants of the structures that may occur. This book is intended for graduate students and researchers in the fields mentioned above. It contains, besides exercises aimed at giving insights, numerous research problems motivated by the developments reported.

Integral closure. --- Algebra. --- Closure, Integral --- Commutative rings --- Ring extensions (Algebra) --- Mathematics --- Mathematical analysis --- Geometry, algebraic. --- Number theory. --- Commutative Rings and Algebras. --- Algebraic Geometry. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry --- Geometry --- Commutative algebra. --- Commutative rings. --- Algebraic geometry. --- Rings (Algebra)

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This monograph is devoted to a new class of non-commutative rings, skew Poincaré–Birkhoff–Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Gröbner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin–Schelter regular algebras, and the noncommutative Zariski cancellation problem. The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.

Associative rings. --- Rings (Algebra). --- Category theory (Mathematics). --- Homological algebra. --- Algorithms. --- Associative Rings and Algebras. --- Category Theory, Homological Algebra. --- Algorism --- Algebra --- Arithmetic --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Foundations --- Ring extensions (Algebra) --- Noncommutative rings. --- Categories (Mathematics) --- Non-commutative rings --- Associative rings --- Extensions of rings (Algebra) --- Associative algebras. --- Algebra, Homological. --- Algebras, Associative