The aim of this work is to study some noncommutative differential operators. We take an algorithmic approach as well as further developing the mathematics, and design and analyse algorithms to solve fundamental problems. We also give applications to differential equations. First we consider how to factor skew polynomials. These are polynomials in a differential or difference operator. Using the...

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions a coefficient by (elementary) matrix-variables over ground ring in not necessary commuting variables. These subalgebras full rings show up noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining precise lift Hilbert Basis Theorem when...

We investigate deformations of skew group algebras arising from the action symmetric on polynomial rings over fields arbitrary characteristic. Over real or complex numbers, Lusztig’s graded affine Hecke algebra and analogs are all isomorphic to Drinfeld algebras, which include symplectic reflection rational Cherednik algebras. prime characteristic, new arise that capture both a disruption also ...

In analogy to cyclic codes, we study linear codes over finite fields obtained from left ideals in a quotient ring of a (non commutative) skew polynomial ring. The paper shows how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is a generalization of the θ-cyclic codes discussed in [1]. However θ-cyclic codes are performant repr...

We describe an attack on the family of Diffie-Hellman and El-Gamal like cryptosystems recently presented at PQ Crypto 2010. We show that the reference hard problem is not hard. 1 Description of the Cryptosystems Skew polynomials are polynomials with a particular noncommutative inner product. Let Fq denote the finite field with q elements, and p be the characteristic of the field. Automorphisms ...

We define a nonassociative generalization of cyclic Azumaya algebras employing skew polynomial rings $D[t;\sigma]$, where $D$ is an algebra constant rank with center $C$ and $\sigma$ automorphism $D$, such that $\sigma|_{C}$ has finite order. The automorphisms these are canonically induced by ring the $D[t;\sigma]$ used in their construction. achieve description inner automorphisms. Results on ...

We prove a new characterization of graded von Neumann regular rings involving the recently introduced class nearly epsilon-strongly rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are regular. More precisely, show algebra LR(E) with coefficients in unital ring R is if and only also both corner skew Laurent polynomial semiprimitive semiprime. ...