Wegen ${\displaystyle {}1\in S}$ ist für ${\displaystyle {}f\in I}$ direkt ${\displaystyle {}1\cdot f\in I}$, also ${\displaystyle {}f\in J}$. Somit umfasst ${\displaystyle {}J}$ das Ideal ${\displaystyle {}I}$ und enthält insbesondere die ${\displaystyle {}0}$. Seien ${\displaystyle {}f_{1},f_{2}\in J}$. Dann gibt es ${\displaystyle {}s_{1},s_{2}\in S}$ mit ${\displaystyle {}s_{1}f_{1},s_{2}f_{2}\in I}$. Da ${\displaystyle {}I}$ ein Ideal ist, gehören auch ${\displaystyle {}s_{1}s_{2}f_{1},s_{1}s_{2}f_{2}\in I}$. Somit ist

${\displaystyle {}s_{1}s_{2}(f_{1}+f_{2})=s_{1}s_{2}f_{1}+s_{1}s_{2}f_{2}\in I\,}$

${\displaystyle {}s_{1}s_{2}\in S}$

${\displaystyle {}f_{1}+f_{2}\in J}$

${\displaystyle {}f\in J}$

${\displaystyle {}r\in R}$

${\displaystyle {}s\in S}$

${\displaystyle {}sf\in I}$

${\displaystyle {}rsf=srf\in I}$

${\displaystyle {}rf\in J}$