Köthe Conjecture

GPT-6 Astra constructs a unital algebra $R=k\oplus A$ over the countable field $$k=\overline{\mathbb F_2},$$ with $I=A$ a nil two-sided ideal, together with a matrix $$W\in M_2(I)$$ that is not nilpotent. The algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$. This formally disproves Krempa's matrix formulation of Köthe's conjecture.

GPT-6 Astra constructs a unital algebra $R=k\oplus A$ over the countable field $$k=\overline{\mathbb F_2},$$ with $I=A$ a nil two-sided ideal, together with a matrix $$W\in M_2(I)$$ that is not nilpotent. The algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$. This formally disproves Krempa's matrix formulation of Köthe's conjecture.