For every finite associative unital ring R, let N(R):=f(x)∈ R[x]| f(a)=0 for every a∈ R be the null ideal of R. The conjecture asserts that N(R) is a two-sided ideal of R[x].

Two-sidedness conjecture for null ideals of finite rings

Werner conjectured that, for every finite ring $R$ , the null polynomials form a two-sided ideal of $R[x]$ . The general assertion was already contradicted by an explicit $4\times4$ upper-triangular example.

Known results

Werner: the property holds for local rings, semisimple rings, matrix rings over commutative rings, and rings of odd order.
Frisch: it holds for upper-triangular and structural matrix rings.

August 26, 2026: counterexamples at all higher levels

A new paper reports counterexamples for every nilpotency level $n\ge5$ and proves the property when radical nilpotency is at most $3$ . A separate preprint gives an explicit finite-ring counterexample and states that GPT-5.6 Sol generated the counterexample and initial proof; the author reports independent verification and simplification.

Current status (as of August 2026): The general conjecture has a claimed counterexample, while the all-level family and the positive range through radical nilpotency $3$ remain unverified in this record.

Two-sidedness conjecture for null ideals of finite rings

Primary source

Additional references

Progress summary

Solutions 0