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The following question is from my exercise sheet in Commutative Algebra and I am not making any sufficient progress on it.

Question: Let $A$ be a ring and let $I$ be a finitely generated two-sided ideal of $A$ such that $I^2 =I$. Show that there exists a central element $e\in A$ such that $e=e^2$ and $I=eA= Ae$.

Using Nakayma Lemma I got that there exists an $r\in A$ such that $rI=0$ and $r= 1+ aI$. But I am not able to show the existence of such an element $e=e^2$ and I.

Can you please help me with this by giving some hints?

  • The word "central" seems redundant assuming your ring $A$ is commutative. You are trying to show that $I$ is principal. You showed that there exists $r \in A$ such that $rI = 0$ and $r = 1 + i$ for $i \in I$. Try substituting $r = 1 + i$ in $rI = 0$. What do you observe? – Haran Feb 12 '23 at 15:44
  • Do you really assume $A$ commutative? if so, "two-sided" and "$eA=Ae$" are also redundant, and https://math.stackexchange.com/questions/297624/use-nakayamas-lemma-to-show-that-i-is-principal-generated-by-an-idempotent?noredirect=1&lq=1 is a duplicate – Anne Bauval Feb 12 '23 at 16:01
  • If you don't assume $A$ commutative, this post might be of interest: https://math.stackexchange.com/questions/1189694 – Anne Bauval Feb 13 '23 at 09:30
  • @AnneBauvel No, A is not commutative here. –  Feb 13 '23 at 13:51

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