Ideal Distributive Law
Distributive Law for Ideals
Given any ideals \(I, J, K\) of a ring \(R\), we have that
\[ I(J + K) = IJ + IK.\]
Proof
\[\begin{align*}
I(J + K) &= I \langle J \cup K \rangle \\
&= \langle \{ im : i \in I , m \in J \cup K \} \rangle \\
&= \langle \{ ij : i \in I, j \in J \} \cup \{ ik : i \in I, k \in K \} \rangle \\
&= \langle \langle\{ ij : i \in I, j \in J \}\rangle \cup \langle\{ ik : i \in I, k \in K \} \rangle\rangle \\
&= \langle\{ ij : i \in I, j \in J \}\rangle + \langle\{ ik : i \in I, k \in K \} \rangle \\
&= IJ + IK
\end{align*}\]