feat: Hopkins–Levitzki theorem #17908
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…endent' into finite_semisimple_module
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| instance : (jacobson R).IsTwoSided := | ||
| ⟨fun b ha ↦ Module.le_comap_jacobson (f := LinearMap.toSpanSingleton R R b) ha⟩ |
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@Vtec234 Here's an easy proof that the Jacobson radical of a ring is a two-sided ideal, using Jacobson radicals of modules. (from Lorenz's Algebra, Volume II)
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Neat! Can it be used to simplify the proof of jacobson_mul_mem_right below?
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Show that over a semiprimary (possibly noncommutative) ring (
R/J(R)is semisimple andJ(R)is nilpotent), Noetherianity is equivalent to Artinianity. As a consequence, an Artinian ring is Noetherian.Ideal.IsTwoSidedclass for quotients of noncommutative rings #17930