pointwiseMulSemiringAction — Mathlib

English

The span of a singleton set equals the ideal generated by its element.

Русский

Замыкание множества с одним элементом образует порожденное им идеал.

LaTeX

$$$\mathrm{span}\,R(\{x\}) = \{ r\cdot x : r\in R\} = \mathrm{span}\,R\{x\}$$$

Lean4

/-- The action on a submodule corresponding to applying the action to every element.

This is available as an instance in the `Pointwise` locale.

This is a stronger version of `Submodule.pointwiseDistribMulAction`. -/
protected def pointwiseMulSemiringAction : MulSemiringAction α (Submodule R A)
    where
  __ := Submodule.pointwiseDistribMulAction
  smul_mul r x y := Submodule.map_mul x y <| MulSemiringAction.toAlgHom R A r
  smul_one r := Submodule.map_one <| MulSemiringAction.toAlgHom R A r

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