$A/m_a \simeq B/m_b$? Where $A,B$ are local rings, $A\subset B$ and $m_a\subset m_b$

Question

I'm currently reading a math book in french so I'm translating everything as I go and also proving the remarks made throughout.

One remark that I haven't been able to prove is: if $A$ and $B$ are local rings with respective maximal ideals $m_a$, $m_b$ such that $A\subset B$ and $m_a=A\cap m_b$, then the residue fields $ A/m_a$ and $B/m_b$ are isomorphic.

I defined $\phi:A/m_a\to B/m_b$ such that $\phi(x+m_a)=x+m_b$. I didn't have any issue proving that its well defined, a morphism and injective. I coudn't prove its surjective though, any hints would be appreciated.

Answer 1

The proposition is wrong, I had mistranslated the remark. As peter a g pointed out in the comments there is a counter example.