TY - BOOK ID - 69097220 TI - Skew pbw extensions : ring and module-theoretic properties, matrix and grobner methods, and applications PY - 2020 SN - 3030533778 3030533786 PB - Cham, Switzerland : Springer, DB - UniCat KW - Associative rings. KW - Rings (Algebra). KW - Category theory (Mathematics). KW - Homological algebra. KW - Algorithms. KW - Associative Rings and Algebras. KW - Category Theory, Homological Algebra. KW - Algorism KW - Algebra KW - Arithmetic KW - Homological algebra KW - Algebra, Abstract KW - Homology theory KW - Category theory (Mathematics) KW - Algebra, Homological KW - Algebra, Universal KW - Group theory KW - Logic, Symbolic and mathematical KW - Topology KW - Functor theory KW - Algebraic rings KW - Ring theory KW - Algebraic fields KW - Rings (Algebra) KW - Foundations KW - Ring extensions (Algebra) KW - Noncommutative rings. KW - Categories (Mathematics) KW - Non-commutative rings KW - Associative rings KW - Extensions of rings (Algebra) KW - Associative algebras. KW - Algebra, Homological. KW - Algebras, Associative UR - https://www.unicat.be/uniCat?func=search&query=sysid:69097220 AB - 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. ER -