Answer to Question #25255 in Abstract Algebra for Tsit Lam

Question

Let R be a left primitive ring. Show that for any nonzero idempotent e ∈ R, the ring A = eRe is also left primitive

Answer 1

Let V be a faithful simpleleft R-module. It suffices to show that U = eV is afaithful simple left A-module. FirstA · U = eRe · eV = eReV ⊆ eV = U, so U is indeed an A-module.Let a = ere ∈ A, where r∈ R. Then ae = ere² = a.If aU = 0, then 0 = aeV = aV implies that a = 0, soAU is faithful. To check that AU is simple, let us show that for0 <> u ∈ U and u'∈ U, we have u' ∈ Au. Note that u, u' ∈ eV implies u = eu, u' = eu'.We have u' = ru for some r ∈ R, so u' = eu' = eru = (ere)u∈ Au, as desired.