18.105 (2014)

Let $R$ be the formal power series algebra in commuting indeterminates $x_1, \dots, x_n, \dots$ over the field with $p$ elements, with $p$ a prime. (Thus its elements are (in general infinite) linear combinations of monomials $x_1^{a_1} \dots x_n^{a_n}$ with $n, a_1, \dots, a_n$ non-negative integers.) Let $I$ be the maximal ideal of $R$ and $J$ the (abstract) ideal generated by all products of two elements of $I$. Is there an ideal $U$ of $R$ such that $U < I$ and $U + J = I$?

If so, then $\text{SL}_3(R)$ and the kernel of the map to $\text{SL}_3(R/U)$ provide an answer to 18.104.