# Ascending chain condition on ideals of free products

In my previous question: M Shahryari (mathoverflow.net/users/29488), Normal Subgroups of Free Products, Normal Subgroups of Free Products (version: 2012-11-28), I asked if a group $A$ has max-n property, is it true that the free product $A\ast \mathbb{Z}$ has also max-n? The answer was NO in that case. Now suppose $F$ is free group of finite rank and $A$ is a group having max-n (maximal condition on normal subgroups). A normal subgroup $N$ of $A\ast F$ is called an ideal if $N\cap A=1$. Is it true that $A\ast F$ has maximal property of ideals?

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Looking at the comments on your previous question, I see that you are interested in groups which are not equationally Noetherian. You might be interested in this question of mine: mathoverflow.net/questions/75784/… . –  HJRW Dec 6 '12 at 14:15
In particular, you might be interested in Denis Osin's answer to that question. –  HJRW Dec 6 '12 at 14:16