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Answer to Question #25255 in Abstract Algebra for Tsit Lam

Question #25255
Let R be a left primitive ring. Show that for any nonzero idempotent e ∈ R, the ring A = eRe is also left primitive
Expert's answer
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.

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