is (A+B)/B=A/B true for A, B modules?

Question

If I have $A$ and $B$ two modules, is the following reasoning true? $(A+B)/B=\{a+b+B : a\in A, b \in B \}=\{a+B : a \in A \}=A/B$

I am just doubting it since it is weirdly similar to the second isomorphism theorem but still not the same thing.

Answer 1

No, consider abelian groups as $\Bbb Z$-modules:

By the second isomorphism theorem your proposition gives $A / B = B / (A \cap B)$

Let $A=\Bbb Z_{12}$, $B = \Bbb Z_3$. Then $A/B = \Bbb Z_4$ but $B/(A \cap B)$ is a quotient group of $\Bbb Z_3$ so it is not equal to $\Bbb Z_6$.