I am studying some algebra during my spare time. In particular I am learning about Noetherian rings. A friend sent me the following excersise, and I am not able to solve it.

Suppose that a ring $A$ is generated by a subring $R$ and a nilpotent element $n$ such that $R+nR=R+Rn$. (Dis)prove:

$R$ left noetherian implies that $A$ is left noetherian.

I believe that the statement above is correct. However I failed in trying to prove that $A$ satisfies the ascending chain condition on ideals, or in proving that every ideal of $A$ is finitely generated. I also thought about using induction as the statement is trivial if $n=0$ but I do not see really how to progress from there on. Moreover I thought about writing $A$ as a ring isomorphic to a quotient of $R[X]$ and then using the Hilbert basis theorem, but I am not convinced.

Would somebody be so kind to shed light on this problem?

Thanks in advance!


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