finding example for modular law if the condition C⊆B is necessary

Question

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.

Do you mean sets, not modules? Do you mean the union, denoted "$\cup$" instead of "+" (meaning ambiguous)? — Newb, Mar 30 '14 at 07:42
Your "For example..." is confusing since you begin your question with "..of module $;A;$." , so it'd seem you need here modules, not simply sets which, in general, we don't know how to sum with each others. — DonAntonio, Mar 30 '14 at 10:30

score 1 · Answer 1 · answered Mar 30 '14 at 10:37

1

Take $\;A:=2\Bbb Z\le \Bbb Z\;,\;\;B=6\Bbb Z\,,\,\,C=4\Bbb Z\,,\,D=10\Bbb Z$ , then:

$$\begin{cases}B\cap(C+D)=B\cap A=B\\{}\\C+(B\cap D)=C+30\Bbb Z=A\end{cases}\;\;\text{and}\;\;A\neq B$$

answered Mar 30 '14 at 10:37

DonAntonio

1 Answers1