Abstract:
After recalling the definition of codes as modules over skew polynomial rings, whose multiplication is defined by using an automorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of their main algebraic and geometric properties. Finally, for module skew codes constructed only with an automorphism, we give some BCH type lower bounds for their minimum distance.

Abstract:
Let $\alpha$ be an endomorphism of a ring $R$. We introduce the notion of weak $\alpha$-skew McCoy rings which are a generalization of the $\alpha$-skew McCoy rings and the weak McCo rings. Some properties of this generalization are established, and connections of properties of a weak $\alpha$-skew McCoy ring $R$ with $n\times n$ upper triangular $T_n(R)$ are investigated. We study relationship between the weak skew McCoy property of a ring $R$ and it's polynomial ring, $R[x]$. Among applications, we show that a number of interesting properties of a weak $\alpha$-skew McCoy ring $R$ such as weak skew McCoy property in a ring $R$.

Abstract:
Let $R=K[x;sigma]$ be a skew polynomial ring over a division ring $K$. We introduce the notion of derivatives of skew polynomial at scalars. An analogous definition of derivatives of commutative polynomials from $K[x]$ as a function of $K[x] ightarrow K[x]$ is not possible in a non-commutative case. This is the reason why we have to define the derivative of a skew polynomial at a scalar. Our definition is based on properties of skew polynomial rings, and it makes possible some useful theorems about them. The main result of this paper is a generalization of polynomial interpolation problem for skewpolynomials. We present conditions under which there exists a uniquepolynomial of a degree less then $n$ which takes prescribed values at given points $x_{i}in K$ ($1leqn$). We also discuss some kind of {sc Silvester-Lagrange} skew polynomial.

Abstract:
Two are the objectives of the present paper. First we study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and that of the local ring RP with respect to a prime ideal P of R and we prove a criterion about the D- simplicity of R in case where R is a 1-dimensional (Krull dimension) finitely generated algebra over a field of characteristic zero and D is a singleton set. The above criterion was quoted without proof in an earlier paper of the author. Second we construct a special class of iterated skew polynomial rings defined with respect to finite sets of derivations of a ring R (not necessarily commutative) commuting to each other. The important thing in this class is that, if R is a commutative ring, then its differential simplicity is the necessary and sufficient condition for the simplicity of the corresponding skew polynomial ring. Key-Words- Derivations, Differentially simple rings, Finitely-generated algebras, Iterated skew polynomial rings, Simple rings.

Abstract:
A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These extend some known results concerning a description of quasi-duo polynomial rings and give a partial answer to the question posed by Lam and Dugas whether right quasi-duo rings are left quasi-duo.

Abstract:
In this note we consider the links of prime ideals of certain skew polynomial rings and prove our main theorem, namely theorem [5], which states the following.Let R be a noetherian ring that is link k-symmetric and let {\sigma} be an automorphism of R.Let S(R) denote the skew polynomial ring R[x,{\sigma}]. Let B be a prime ideal of S(R) that is extended from R. Then, for a prime ideal D of S(R),there is a link D\rightarrowB in the ring S(R) implies that D is an extended prime ideal of S(R).

Abstract:
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 polynomial rings, with $r=0$, as such Artamonov shows in his work. We use this result for giving some more general results for skew $PBW$ extensions, using filtred-graded techniques. Finally, we use localization for characterize some class the endomorphisms and automorphisms for skew $PBW$ extensions and skew quantum polynomials over Ore domains.

Abstract:
We study the Hochschild homology of the iterated skew polynomial rings introduced by D. Jordan in ``A simple localization of the quantized Weyl algebra''. First, we obtain a complex, smaller than the canonical one of Hochschild, given the homology of such an algebra, and then, we study this complex in order to compute the homology of some families of algebras. In particular we compute the homology of some quantum groups, in the generic case.

Abstract:
In this article, we study the relationship between left (right) zip property of and skew polynomial extension over , using the skew versions of Armendariz rings.

Abstract:
Skew polynomial rings were used to construct finite semifields by Petit in 1966, following from a construction of Ore and Jacobson of associative division algebras. In 1989 Jha and Johnson constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in 2008 and implicitly in recent work by Dempwolff.