The Maximal Integral Domain Generated By A Commutative Ring

Let $\Bbbk$ be an algebraically closed field of characteristic zero and let $f$ be a nonconstant polynomial in finitely many freely noncommuting variables. We prove the asymptotic Makar-Limanov rank conjecture, namely that the normalized rank of a value of $f$ can be made arbitrarily small by choosing matrices over $\Bbbk$ of finite size.

math.RA↗

arXiv2026

Nilpotent Product Probability of Finite Rings

For a finite ring $R$, we investigate the probability that the product of two randomly chosen elements in $R$ is nilpotent, which we call the nilpotent product probability (NPP) of $R$ and denote by $P_{nil}(R)$. We derive bounds for $P_{nil}(R)$ that depend on the structure of $R$, and prove that among all commutative rings with identity, the class of local rings with residue field $\mathbb{Z}_{2}$ attains the maximum value of NPP, which is equal to $3/4$. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of $R$ and NPP of some subrings of the matrix ring over $R$.

math.RA↗

arXiv2026

Diagonalizable modular derivations on bigraded Poisson spaces and Koszul duality

For two bigraded spaces $k^{m||n}$ and $k^{n\wedge m}$ equipped with appropriate quadratic Poisson structures, Koszul duality identifies the mixed Poisson differential calculi and, in the unimodular case, their Poisson cohomologies as Batalin--Vilkovisky algebras. In the current paper we prove that Koszul duality induces an isomorphism of Batalin--Vilkovisky algebras between their Poisson cohomologies when the modular derivation of one algebra is diagonalizable, including non-unimodular cases.

math.RA↗