instCoeOutSubmodule — Mathlib

English

There is a canonical CoeOut from FractionalIdeal to Submodule defined by the forgetful map to the underlying submodule.

Русский

Существует канонический переход (коeOut) от FractionalIdeal к Submodule через Forgotten-слой подмодуля.

LaTeX

$$$\\text{instOut}\: (FractionalIdeal\\,S\\,P) \\Rightarrow Submodule\\,R\\,P$$$

Lean4

/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.

This coercion is typically called `coeToSubmodule` in lemma names
(or `coe` when the coercion is clear from the context),
not to be confused with `IsLocalization.coeSubmodule : Ideal R → Submodule R P`
(which we use to define `coe : Ideal R → FractionalIdeal S P`).
-/
instance : CoeOut (FractionalIdeal S P) (Submodule R P) :=
  ⟨coeToSubmodule⟩

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