Kourovka Notebook Problem 18.105

Definitions and mathematical statement

Let $p$ be a prime, and let ${\mathbb{F}}_p$ be the field with $p$ elements. Write \[ M=\left\{(a_i)_{i\geq 1}:a_i\in{\mathbb{Z}}_{\geq 0},\ a_i=0\text{ for all but finitely many }i\right\}, \qquad x^a=\prod_{i\geq 1}x_i^{a_i}. \] Following the explicit description in the supplied source, define the commutative formal power series algebra \[ R=\left\{\sum_{a\in M}c_a x^a:c_a\in{\mathbb{F}}_p\right\}, \] with no restriction on the number of nonzero coefficients. Addition is coefficientwise, and multiplication is given by \[ \left(\sum_{a\in M}c_a x^a\right)\left(\sum_{b\in M}d_b x^b\right) =\sum_{e\in M}\left(\sum_{a+b=e}c_a d_b\right)x^e. \] For each fixed $e$, the inner sum is finite. Let \[ I=\left\{\sum_{a\in M}c_a x^a\in R:c_0=0\right\} \] be the maximal ideal of series with zero constant coefficient. Define the abstract ideal generated by products of two elements of $I$ by \[ J=I^2=\left\{\sum_{k=1}^{m}f_k g_k:m\in{\mathbb{Z}}_{\geq 0},\ f_k,g_k\in I\right\}. \] Here the sum is finite, and the empty sum is zero; no closure operation is included in the definition of $J$.

Question (John S. Wilson, Kourovka Notebook 18.105). For a prime $p$ and the algebra $R$ above, does there exist an ideal $U$ of $R$ such that \[ U\subsetneq I \qquad\text{and}\qquad U+J=I? \] The prime is an arbitrary parameter; the ideal $U$, if it exists, may depend on $p$.

The source additionally states that an affirmative answer makes $\mathrm{SL}_3(R)$ and the kernel of the reduction homomorphism \[ \mathrm{SL}_3(R)\longrightarrow\mathrm{SL}_3(R/U) \] provide an answer to the preceding question. Here $\mathrm{SL}_3(A)$ denotes the group of $3\times3$ matrices of determinant one over a commutative ring $A$. The preceding question is not included in the supplied record, so the intended group-theoretic consequence cannot be stated completely from this evidence.

Short English statement

For the algebra of arbitrary formal series in countably many commuting variables over ${\mathbb{F}}_p$, let $I$ be the ideal of series with zero constant term. Wilson asks whether a proper ideal $U\subsetneq I$ can satisfy $U+I^2=I$, where $I^2$ consists of finite sums of products of elements of $I$. The source's further application refers to an unavailable preceding question.

Sources

• Local UnsolvedMath registry, permanent ID $701533$, exact source problem code catalogue entry, title Kourovka Notebook Problem 18.105. The supplied record has upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82 and records the status as open; this is supplied registry metadata, not an independently verified mathematical or literature assessment.
• The supplied attribution is to John S. Wilson, proposed in 2014. The background identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), and the 21st issue edition (2026), arXiv:1401.0300v48, original problem number 18.105. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of the Issues 1--21 collection. These bibliographic details are reproduced from the record and have not been independently checked.