jacobson_smul_lt_top — Mathlib

English

If I ≤ jacobson R, then jacobson(R/I) equals the image of jacobson R under the quotient map by I.

Русский

Если I ≤ jacobson(R), то jacobson(R/I) равен образу jacobson(R) по тождественному каналу: через факторацию по I.

LaTeX

$$$\\operatorname{jacobson}(R/I) = \\operatorname{Submodule.map}(\\operatorname{Ideal.Quotient.mk} I).toSemilinearMap(\\operatorname{jacobson}(R)).$$$

Lean4

theorem jacobson_smul_lt_top [Nontrivial M] [IsCoatomic (Submodule R M)] (N : Submodule R M) :
    Ring.jacobson R • N < ⊤ :=
  ((smul_mono_right _ le_top).trans <| Ring.jacobson_smul_top_le R M).trans_lt (Module.jacobson_lt_top R M)

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