Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal? I have a question concerning the following local ring:
$$R=K[X_1,...X_n,...]/(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...).$$

Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal? 

As for any ideal of the form $(\bar X_n)$ which is a nilpotent ideal the answer is trivially in affirmative.
 Thanks for any suggestion!
 A: Let $J$ be the ideal $(X_1, X_2^2-X_1,...,X_{n+1}^2-X_n,...)$ of
$K[X_1,X_2,...]$, and let $x_i=X_i+J$ for all $i$. Thus, for each
$i<j$ there exists $n$ such that $x_i=x_j^n\quad (*)$.
It follows that
each term $x_{i_1}^{n_1}x_{i_2}^{n_2}\cdots x_{i_r}^{n_r}$ can be
written as $x_k^m$ for $i_1,i_2,...,i_r\leq k$. Hence, if $f\in
R$ there exists $k$ such that
$$f=k_1x_k^{n_1}+k_2x_k^{n_2}+\cdots+k_{r}x_k^{n_r}$$ for ${n_1}<{n_2}<\cdots<{n_r}$,
and so
$f=x_k^{n_1}(k_1+k_2x_k^{n_2-n_1}+k_3x_k^{n_3-n_1}+\cdots+k_{r}x_k^{n_r-n_1})$,
where $k_1+k_2x_k^{n_2-n_1}+k_3x_k^{n_3-n_1}+\cdots+k_{r}x_k^{n_r-n_1}$ is a
unit in $R$.
It follows if $I$ is an ideal of $R$ such that
$f\in I$ then $x_k^{n_1}\in I$. Therefore, if $I$ is an ideal of
$R$ then there exists a family $\{x_\alpha\}_{\alpha\in \Gamma}$
and $\{n_\alpha\}_{\alpha\in\Gamma}\subseteq\mathbf{N}$ such
that $I=(x_\alpha^{n_\alpha})_{\alpha\in\Gamma}$. Now, if
$\Gamma$ is a finite set then $I$ is nilpotent, and if $\Gamma$
is an infinite set then
$I=(x_\beta^{n_\beta})+(x_\alpha^{n_\alpha})_{\alpha\in\Gamma-\{\beta\}}$,
and since $\Gamma-\{\beta\}$ is infinite,
$(x_\alpha^{n_\alpha})_{\alpha\in \Gamma-\{\beta\}}$ is
idempotent ideal of $R$ by $(*)$ property. (I hope this helps you.)
