Notation: $R/A$ is a factor ring.
I am having a problem understanding the final part of the textbook proof. It refers to an exercise in the textbook that I am unsure how it applies. I understand the other direction so I will only include the one direction I do not understand. Thank you for your time. Note that the proof below is my interpretation.
Proof: $R/A$ is a field $\Rightarrow$ $A$ is maximal
Suppose $R/A$ is a field, $B$ is an ideal of $R$ such that $A \subseteq B \subseteq R$
Let $\alpha \in B$ and $\alpha \not\in A$. Then $\alpha+A$ is a non zero element of $R/A$ since $$\alpha \not\in A \ \therefore \exists \beta \in A : (\alpha+A)(\beta + B) = \alpha\beta + A = 1 + A$$ (since $R/A$ is a field (multiplicative inverse)).
Since $A \subseteq B \subseteq R$, and $\alpha \in B$, then $\alpha\beta \in B$ since $$1 + A = \alpha\beta + A \Rightarrow 1-\alpha\beta \in A \subset B$$
So $$(1 + \alpha\beta) - \alpha\beta + B \Rightarrow 1 \in B$$
Hence $B = R$
This proves $A$ is maximal
I dont understand the last part of the prove. Since $\beta \in A$, and $A \subseteq B$, and $\alpha \in A$, then doesnt $\alpha\beta + B = 0? $ And from my understanding, maximal means if $A,B$ are ideals of $R$ such that $A \subseteq B \subseteq R$, then $A = B$ or $B = R$, what approach would we take to show $A = B$ ? I don't think its sufficent to show one part of the or statement. Thank you again