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I drool over a proof of a theorem called the correspondence theorem. That's the issue :

Let $A$ be a ring and let $I$ be an ideal of $A$. Then there exists a bijection between the set of ideals of $A$ containing $I$ and the set of ideals of $A / I$ . $i.e$ : There exists a bijection $\lambda : \{ J \,/\, J\,$ ideal $\,of\,A\,; I \subset J \} \to \{K\, / \,K\,$ ideal $\, of\,A/I \}$

Can someone please explain me how to define such a bijection. And thank you in advance.

Robert Cardona
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Mohammed Sabak
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1 Answers1

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Give a name to the canonical homomorphism. Say, $\phi\colon A\to A/I$. For an ideal $J$ of $A$ the image in $A/I$, viz, $\phi(J)$, because of surjectivity of $\phi$, is an ideal of $A/I$. This provides the bijection $\lambda$. Inverse image of an ideal is always an ideal under any ring homomorphism.

P Vanchinathan
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