A full idempotent with a basic corner in a semisimple ring #
Every semisimple ring R has a full idempotent e, one with TwoSidedIdeal.span {e} = ⊤,
whose corner ring eRe is a finite product of division rings. By Artin–Wedderburn,
R ≅ ∏ᵢ Matₙᵢ(Dᵢ); taking the matrix unit E₁₁ in every block gives an idempotent whose corner
is ∏ᵢ Dᵢ, and it is full because each E₁₁ is full in its block.
A full idempotent makes R Morita equivalent to its corner eRe
(IsIdempotentElem.isMoritaEquivalent_corner), and a product of division rings is the
semisimple case of a basic ring (TauCeti.isBasic_iff_pi_divisionRing). So this is the selection
step of the reduction of a finite-dimensional algebra A to a basic one: an idempotent chosen
here in the semisimple quotient A ⧸ Ring.jacobson A lifts to a full idempotent of A, whose
corner is then basic. In module-theoretic terms, the left ideal R e constructed here is a sum of
one copy of each simple R-module, matching the classical description of the basic algebra as
the endomorphism ring of one indecomposable projective per simple module.
Main results #
IsSemisimpleRing.exists_isIdempotentElem_full_corner_ringEquiv_pi_divisionRing: a semisimple ring has a full idempotent whose corner is a finite product of division rings.
References #
- I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras I, CUP (2006), Section I.6.
- T. Y. Lam, Lectures on Modules and Rings, GTM 189, Springer (1999), §18B.
A semisimple ring has a full idempotent whose corner is a finite product of division
rings. Writing R ≅ ∏ᵢ Matₙᵢ(Dᵢ) by Artin–Wedderburn, the idempotent is the matrix unit E₁₁
of every block: its corner is ∏ᵢ Dᵢ, and it generates R as a two-sided ideal.