Finding the image of a ring homomorphism

Question

Let $A$ be a subring and $B$ be an ideal of a ring $R$. I'm trying to prove the second isomorphism theorem for ring, which state that $$A/(A \cap B) \cong (A+B)/B$$

So I have defined a map $$f: A \longrightarrow R/B$$ $$f(a) = a + B$$ clearly $f$ is a ring homomorphism. Since $ker(f) = A \cap B$ , the result follow once I show that $im(f) =(A+B)/B$ by the first isomorphism theorem.

So \begin{align} im(f) & = \{f(a) : a \in A\} & (1)\\ & = \{a + B : a \in A\} & (2) \\ & = \{a + b + B : a \in A, b \in B\} & (3) \\ & = (A + B)/B & (4) \end{align}

question:

1. is this valid? (specifically (2) to (3))

2. is it true that $A/B \cong (A + B)/B$

Answer 1

1. yes, the proof works exactly like that.
2. well, yes: the notation "$A/B$" is only really used if $B\subset A$, i.e. if $A$ is a subring of $B$ or an ideal in $B$. In that case $A+B=A$, so your statement is trivially true.