Finite nonassociative algebras obtained from skew polynomials and possible applications to $(f,σ,δ)$-codes

Abstract

Let $S$ be a unital ring, $S[t;σ,δ]$ a skew polynomial ring where $σ$ is an injective endomorphism and $δ$ a left $σ$-derivation, and suppose $f\in S[t;σ,δ]$ has degree $m$ and an invertible leading coefficient. Using right division by $f$ to define the multiplication, we obtain unital nonassociative algebras $S_f$ on the set of skew polynomials in $S[t;σ,δ]$ of degree less than $m$. We study the structure of these algebras. When $S$ is a Galois ring and $f$ base irreducible, these algebras yield families of finite unital nonassociative rings $A$, whose set of (left or right) zero divisors has the form $pA$ for some prime $p$. For reducible $f$, the $S_f$ can be employed both to design linear $(f,σ,δ)$-codes over unital rings and to study their behaviour.