Narrow Results

Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota–Baxter operator. In applying the general framework to univariate polynomials, one is led to the integro–differential analogs of the classical Weyl algebra. These are analyzed in terms of skew polynomial rings and noncommutative Gröbner bases.

Given an integral domain $D$ with quotient field $K$, the study of the ring of integer-valued polynomials $\mathrm{\text{Int}}(D)=\{f\in K[X]|f(a)\in D\text{for all}a\in D\}$ has attracted a lot of attention over the past decades. Recently, Werner has extended this study to the situation of skew polynomials. To be more precise, if $\sigma$ is an automorphism of $K$, one may consider the set $\mathrm{\text{Int}}(D,\sigma )=\{f\in K[X,\sigma ]|f(a)\in D\text{for all}a\in D\}$, where $K[X,\sigma ]$ is the skew polynomial ring and $f(a)$ is a “suitable” evaluation of $f$ at $a$. For example, he gave sufficient conditions for $\mathrm{\text{Int}}(D,\sigma )$ to be a ring and study some of its properties. In this paper, we extend the study to the situation of the skew polynomial ring $K[X,\sigma ,\delta ]$ with a suitable evaluation, where $\delta$ is a $\sigma$-derivation. Moreover we prove, for example, that if $\sigma$ is of finite order and $D$ is a Dedekind domain with finite residue fields such that $\mathrm{\text{Int}}(D,\sigma )$ is a ring, then $\mathrm{\text{Int}}(D,\sigma )$ is non-Noetherian.

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,\sigma ,\delta ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a\in D$ and $f\in D[x,\sigma ,\delta ]$ satisfy $f(a)=a$, then $f^{\circ n}(a)=a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.

In this paper, we focus on the algebraic structure of left negacyclic codes of length $p^s$ over the finite non-commutative chain ring $\frac{{𝔽}_{p^m}[u,𝜃]}{〈u^2〉}$ where $𝜃$ is an automorphism on ${𝔽}_{p^m}$. After that, the number of codewords of all left negacyclic codes is obtained. For each left negacyclic code, we also obtain the structure of its right dual code. In the remaining result, the number of distinct left negacyclic codes is given. Finally, a one-to-one correspondence between left negacyclic and left $\gamma$-constacyclic codes of length $p^s$ over $\frac{{𝔽}_{p^m}[u,𝜃]}{〈u^2〉}$ is constructed via ring isomorphism, which carries over the results regarding left negacyclic codes corresponding to left $\gamma$-constacyclic codes of length $p^s$ over $\frac{{𝔽}_{p^m}[u,𝜃]}{〈u^2〉}$ where $\gamma$ is a nonzero element of the field ${𝔽}_{p^m}$ such that $𝜃(\gamma )=\gamma$.