English
Equality of Jacobson radicals of bottom ideals respects the matrix construction: (⊥ : TwoSidedIdeal R).jacobson.matrix n = (⊥ : TwoSidedIdeal (Matrix n n R)).jacobson.
Русский
Равенство радикала Якобсона для нижних идеалов сохраняется при построении матриц: (⊥).jacobson.matrix n = (⊥)matrixn(R).jacobson.
LaTeX
$$$$ (\\perp)^{\\mathrm{jacobson}}_n = (\\perp^{\\mathrm{jacobson}})_{\\mathrm{matrix}}^{(n)}. $$$$
Lean4
theorem matrix_jacobson_bot : (⊥ : TwoSidedIdeal R).jacobson.matrix n = (⊥ : TwoSidedIdeal (Matrix n n R)).jacobson :=
matrix_bot n (R := R) ▸ (jacobson_matrix _).symm