1-1 correspondence theorem

Question

Here is the correspondence theorem stated as follows:

Let $A$ be an Ideal of ring $R$.There is 1-1 correspondence between Ideals of $B$ containing $A$ and ideals of $R/A$. I have read the proof but didn't understand what it actually means and I can't understand things like show that if $B$ is an ideal of $R$ containing $A$ then $B/A$ is ideal containing $A$

Is there some simpler and intutive way of thinking how the proof of the theorem proceeds ...

Answer 1

Let $\;B\;$ be an ideal of $\;R\;$ containing $\;A\;$, and now define :

$$B/A:= \{r+a\in R/A\;\;;\;\;r\in B\}$$

Prove that $\;B/A\;$ as defined above is an ideal of the factor ring $\;R/A\;$ . Important: don't forget that $\;A\le B\;$ !

Now, let $\;\overline B\le R/A\;$ (an ideal of $\;R/A\;$ ), and define

$$B:=\{r\in R\;\;:\;\;r+A\in\overline B\}$$

Prove that $\;B\;$ as defined above is an ideal of $\;R\;$ containing $\;A\;$ .

There you have the main part of the very important Correspondence Theorem for Rings, which allows us to write any ideal of the factor ring $\;R/A\;$ in the form $\;B/A\;$ , for some ideal $\;A\le B\le R\;$ .

You could also prove that both maps determined by the definitions above are inverse to each other, and that the correspondence respects the indexes:

$$[R:B]=\left[R/A:B/A\right]$$