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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Noncommutative algebraic geometry is a branch of mathematics that studies the geometric properties of non-commutative algebraic objects such as rings. Traditional algebraic geometry relies on the assumption that the product two things is independent of their order: a times b equals b times a. In noncommutative algebraic geometry, this assumption is dropped. This provides a setting which is better adapted to the study of certain phenomena such as those coming from quantum physics. It also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups.