English
The quotient-like construction for submodules, I / J, forms a valid divisor structure with natural operations such as zero, addition, and scalar actions.
Русский
Конструкция I / J подобна делению подмодулей, образуя деление с operasiями нуля, сложения и скалярного действия.
LaTeX
$$$\text{Div} (\mathrm{Submodule}\; R A) = \{ \text{I,J : Submodule } R A \mid \text{множество} \} $$$
Lean4
/-- R-submodules of the R-algebra A are a module over `Set A`. -/
noncomputable instance moduleSet : Module (SetSemiring A) (Submodule R A)
where
smul s P := span R (SetSemiring.down s) * P
smul_add _ _ _ := mul_add _ _ _
add_smul s t P := by simp_rw [HSMul.hSMul, SetSemiring.down_add, span_union, sup_mul, add_eq_sup]
mul_smul s t P := by simp_rw [HSMul.hSMul, SetSemiring.down_mul, ← mul_assoc, span_mul_span]
one_smul P := by simp_rw [HSMul.hSMul, SetSemiring.down_one, ← one_eq_span_one_set, one_mul]
zero_smul P := by simp_rw [HSMul.hSMul, SetSemiring.down_zero, span_empty, bot_mul, bot_eq_zero]
smul_zero _ := mul_bot _
