Polynomial-extension preservation problem for the strong finite type property

For every commutative ring $R$ , if every ideal $I\subseteq R$ contains a finitely generated ideal $J\subseteq I$ and an integer $n\ge 1$ such that $I^n\subseteq J$ , then every ideal of the polynomial ring $R[X]$ contains a finitely generated ideal $K$ and an integer $m\ge 1$ such that $L^m\subseteq K$ . Equivalently: does the strong finite type property of $R$ imply the strong finite type property of $R[X]$ ?

References

Primary source

arXiv

Additional references

Polynomial extensions do not preserve the strong finite type property — arXiv

commutative-algebra ring-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed preprint claims a counterexample, so the preservation conjecture is now reported false in general but has not been independently confirmed.

The problem asks whether the strong finite type property passes from a ring $R$ to its polynomial ring $R[X]$ . Earlier literature described this as open, while a newer preprint claims to settle it negatively.

Known results

Earlier work proves only the weaker statement that if an ideal $I$ is strong finite type, then its extension $I[X]$ is strong finite type; this does not establish preservation of the ring property.

August 2026 counterexample

On August 26, 2026, the preprint Polynomial extensions do not preserve the strong finite type property reported a ring $R$ with the strong finite type property whose polynomial ring $R[X]$ lacks it. If correct, this disproves preservation in general; the report is unrefereed and remains unverified.

Current status (as of August 2026): A preprint claims the general preservation statement is false via a counterexample, but that claim has not been independently verified.

Sources

No solutions have been posted yet.