B,C and D are submodule of module A.
$B\cap (C+D)=C+(B\cap D)$ ( modular law)
i am trying to find example the condition $C \subseteq B$ is necessary for modular law.
For example if i take $B=\{1,2\}$ and $C=\{1,3\}$ and $D=\{3,4\}$
and then $(\{1,2\}\cap(\{1,3\}+\{3,4\})) \neq (\{1,3\}+(\{1,2\}\cap\{3,4\})$
Is it true ? Can i choose this elements for $B$, $C$ and $D$ like that ? any help would be appreciated . + is not instead of U.
Take $\;A:=2\Bbb Z\le \Bbb Z\;,\;\;B=6\Bbb Z\,,\,\,C=4\Bbb Z\,,\,D=10\Bbb Z$ , then:
$$\begin{cases}B\cap(C+D)=B\cap A=B\\{}\\C+(B\cap D)=C+30\Bbb Z=A\end{cases}\;\;\text{and}\;\;A\neq B$$