Thoughts on Eggert's Conjecture

zbMATH Open1303.13004arXiv1206.0326MaRDI QIDQ5168093

George M. Bergman

Publication date: 3 July 2014

Abstract: Eggert's Conjecture says that if R is a finite-dimensional nilpotent commutative algebra over a perfect field F of characteristic p, and R^{(p)} is the image of the p-th power map on R, then dim_F R geq p dim_F R^{(p)}. Whether this very elementary statement is true is not known. We examine heuristic evidence for this conjecture, versions of the conjecture that are not limited to positive characteristic and/or to commutative R, consequences the conjecture would have for semigroups, and examples that give equality in the conjectured inequality. We pose several related questions, and briefly survey the literature on the subject.

Full work available at URL: https://arxiv.org/abs/1206.0326

zbMATH Keywords

finite abelian semigroupsFrobenius map on a finite-dimensional nilpotent commutative algebra

Mathematics Subject Classification ID

Finite rings and finite-dimensional associative algebras (16P10) Commutative semigroups (20M14) Ordinary and skew polynomial rings and semigroup rings (16S36) Nil and nilpotent radicals, sets, ideals, associative rings (16N40) Semigroup rings, multiplicative semigroups of rings (20M25) Characteristic (p) methods (Frobenius endomorphism) and reduction to characteristic (p); tight closure (13A35) Commutative Artinian rings and modules, finite-dimensional algebras (13E10) Local rings and semilocal rings (13H99)

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