I am making a proof and at a certain point I need to use the following, but I am not sure that it is true:
Let $M=M_1\oplus M_2$ be a module over a $K$-algebra $A$, with $M_1,M_2\leq M$. If $N\le M$, then $$\frac{M}{N} \simeq \frac{M_1}{N\cap M_1}\oplus \frac{M_2}{N\cap M_2}.$$
I tried to prove it in a natural way, but I'm missing the last part:
I consider the obvious map $$\phi: m = m_1 + m_2\in M \mapsto (m_1+N\cap M_2, m_2 + N\cap M_2).$$
This map is onto because if I take a pair $(m_1 + N\cap M_1, m_2 + N\cap M_2)$ then it suffices to take $m = m_1 + m_2$ as a pre-image.
As regards the kernel, we have that $m=m_1 + m_2\in Ker\phi \Rightarrow m_1\in N, m_2\in N \Rightarrow m\in N$.
I'm not sure that this holds because I'm not able to prove the converse: if $m= m_1 + m_2 \in N$, how can I prove that both the component are in $N$? It does not seem true, or at least I can not see any reason why it should be.
In case it is not true, what can I say, in general, about the quotient of an internal direct sum?
Thanks everybody in advance!
Best Answer
It's not true. Consider a field $K$, and the vector space $M=K\oplus K$ and take $N=\{(a,a):a\in K\}$. Then $N\cap M_j=0$ for $j=1$, $2$.