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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Oystein Ore, is a special type of a ring extension whose properties are relatively well understood. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.The Weyl algebras are Ore extensions, with R any a commutative polynomial ring, the identity ring endomorphism, and the polynomial derivative.