Let $R$ be a commutative ring and $a,b \in R$. Is it true that $b(R/(a))=((a)+(b))/(a)$?
I think $b(R/(a))=(b)/(a)$. But is there anything wrong in this “proof” that $b(R/(a))=(b)/(a)$?
Proof: $x \in b(R/(a)) \iff x=b\overline{r} = \overline{rb}$ for some $r \in R \iff x \in (b)/(a)$
Notation: $\overline{r} = \pi(r)$ where $\pi: R \to R/(a)$ is the quotient
In one of the steps of a proof to another problem I saw the identity $b(R/(a))=((a)+(b))/(a)$. So I’m quite sure my “proof” is wrong. Can someone provide a counterexample / explain why it is wrong? Thanks.