Wedderburn–Artin theorem

The Wedderburn–Artin theorem classifies semisimple Artinian rings as finite products of full matrix rings over division rings. It also characterizes these rings through their module decompositions.

Semisimple ring: A ring whose modules have no nontrivial extensions between simple constituents; equivalently, its regular module is semisimple. Semisimplicity is the central hypothesis that makes the ring decompose into simple pieces.
Regular module: A ring regarded as a module over itself, usually by left or right multiplication. Its semisimple decomposition exposes the matrix factors in the theorem.
Joseph Wedderburn: A Scottish-American mathematician whose work established foundational results in ring theory and algebra. His 1908 work gave a central early form of the structure theorem.
Representation theory: The study of mathematical structures through their actions on vector spaces or modules. Semisimple ring decompositions reduce representations to simple components.
Jacobson radical: The intersection of the maximal left ideals of a ring, equivalently the annihilator of all simple left modules. A ring is semisimple precisely when its Jacobson radical vanishes under the usual hypotheses.
Artinian ring: A ring satisfying the descending chain condition on ideals. The theorem’s finiteness condition prevents infinitely many simple components.
Artin–Wedderburn decomposition: An isomorphism expressing a semisimple Artinian ring as a finite product of full matrix rings over division rings. This is the explicit structural form asserted by the theorem.
Emil Artin: A mathematician whose work shaped modern algebra, number theory, and the theory of rings. His later formulation extended the theorem beyond finite-dimensional algebras.
Group algebra: An algebra formed from formal linear combinations of elements of a group, with multiplication induced by the group law. Under suitable characteristic conditions, group algebras are semisimple and admit this decomposition.
Artin–Wedderburn theorem for algebras over an algebraically closed field: A specialization in which finite-dimensional division algebras over the field reduce to the field itself. It sharpens the factors to ordinary matrix algebras when the coefficient field is algebraically closed.

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