The ascending chain condition for principal left or right ideals of skew generalized power series rings

Automorphisms for skew PBW extensions and skew quantum polynomial rings

In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We use Artamonov's works as reference for getting the principal results about automorphisms for generic skew $PBW$ extensions and generic skew quantum polynomials. In general, if we have an endomorphism on a generic skew $PBW$ extension and there are some $x_i,x_j,x_u$ such that the endomorphism is not zero on this elements and the principal coefficients are invertible, then endomorphism act over $x_i$ as $a_ix_i$ for some $a_i$ in the ring of coefficients. Of course, this is valid for quantum polyno…

Characterizations and properties of principal $(f, σ, δ)$-codes over rings

Let $A$ be a ring with identity, $σ$ a ring endomorphism of $A$ that maps the identity to itself, $δ$ a $σ$-derivation of $A$, and consider the skew-polynomial ring $A[X;σ,δ]$. When $A$ is a finite field, a Galois ring, or a general ring, some fairly recent literature used $A[X;σ,δ]$ to construct new interesting codes (e.g. skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g. cyclic and constacyclic linear codes). This paper presents results concerning {\it principal} $(f, σ, δ)$-codes over a ring $A$, where $f\in A[X;σ,δ]$ is monic. We…

Noncrossed Product Matrix Subrings and Ideals of Graded Rings

We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced …

利用非结合代数对斜多项式循环码进行到等距和等价的分类

Using nonassociative algebras to classify skew polycyclic codes up to isometry and equivalence

利用非结合代数对斜多项式循环码进行等距和等价分类，通过同构减少分类类别的数量并消除长度限制。

Ore extensions of principally quasi-Baer rings

Let $R$ be a ring and $(σ,δ)$ a quasi-derivation of $R$. In this paper, we show that if $R$ is an $(σ,δ)$-skew Armendariz ring and satisfies the condition $(\mathcal{C_σ})$, then $R$ is right p.q.-Baer if and only if the Ore extension $R[x;σ,δ]$ is right p.q.-Baer. As a consequence we obtain a generalization of \cite{hong/2000}.

$ω$-Euclidean domain and skew Laurent series rings

In this paper we proved that if $R$ is right $ω$-Euclidean domain, then skew Laurent formal series ring is right $ω$-Euclidean domain. We also showed that if $R$ is a right $ω$-Euclidean domain with multiplicative norm, then skew Laurent formal series ring is a right principal ideal domain. In addition, we proved that if $R$ is a noncommutative $ω$-Euclidean domain with a multiplicative norm, then $R$ and skew Laurent formal series ring is a ring with elementary reduction of matrices.