Algebraic aspects of general free skew extensions of rings
Vitor O. Ferreira, Érica Z. Fornaroli, Javier Sánchez
TL;DR
This work develops a comprehensive algebraic framework for general free skew extensions of rings, extending Ore-like multivariate skew polynomial constructions to noncommuting indeterminates with coefficient maps $\sigma$ and $oldsymbol{ abla}$. It connects skew free extensions to universal rings on bimodules, via a tight correspondence with tensor rings and universal $R$-rings, and proves existence, uniqueness, and a universal property. The paper identifies megainjectivity of $\sigma$ as a key condition ensuring domain-ness and introduces a degree-function perspective, linking regularity to a graded-structure property. It further develops a skew series ring under locally nilpotent derivations to accommodate infinite sums and establishes a primeness criterion in upper-triangular settings, with reductions to triangularizable forms. Overall, the results unify and extend the theory of multivariate skew polynomials and Ore extensions, with implications for fraction rings, semiprimitivity, and structural analysis of noncommutative polynomial rings.
Abstract
We consider skew free extensions of rings, also known as free multivariate skew polynomial rings, and explore some of the algebraic aspects of this construction. We give different characterizations of such rings and present conditions for such a ring to be a domain, to be embeddable in a series ring and to be prime.