Noncommutative Algebra
Noncommutative algebra examines algebraic structures where multiplication does not commute, the setting for matrix products, group rings, and division algebras. This MIT OpenCourseWare graduate course covers modules over noncommutative rings, the Wedderburn theory of semisimple algebras, the Jacobson radical, Morita equivalence, and central simple algebras, with applications reaching into representation theory, quantum mechanics, and number theory. Materials include lecture notes and problem sets drawn from MIT's mathematics curriculum, letting a student work through the theory at their own pace. The course assumes a solid background in linear algebra and abstract algebra, including familiarity with rings, fields, and group theory, before tackling how classical structure theorems break down and get replaced once commutativity is dropped. As with other MIT OCW offerings, all materials are free to access under a Creative Commons license, with no certificate offered.