In this paper we generalize coding theory of cyclic codes over finite fields to skew polynomial rings over finite rings. Codes that are principal ideals in quotient rings of skew polynomial rings by two sided ideals are studied. Next we consider skew codes of endomorphism type and derivation type. And we give some examples. Mathematics Subject Classification: Primary 94B60; Secondary 94B15, 16D25

Let K be a field of characteristic p > 0. It is proved that the group Autord(D(Ln)) of order preserving automorphisms of the ring D(Ln) of differential operators on a Laurent polynomial algebra Ln := K[x ±1 1 , . . . , x n ] is isomorphic to a skew direct product of groups Z n p⋊AutK(Ln) where Zp is the ring of p-adic integers. Moreover, the group Autord(D(Ln)) is found explicitly. Similarly, A...

Suppose that G is a simple graph. We prove that if G contains a small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(G) of an arbitrary orientation G of G and the minors of A(G). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the m...

Huber, Krokhin, and Powell (2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain, and Huber and Krokhin (2014) showed the oracle tractability of minimization of skew-bisubmodular functions. Fujishige, Tanigawa, and Yoshida (2014) also showed a min-...

We characterize skew polynomial rings and power series that are reduced right or left Archimedean.

We apply new techniques to compute Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets on polyvector fields. We obtain as consequences some conditions under which brackets are always 0, and show that the Hochs...

We consider separably closed fields of characteristic p > 0 and fixed Imperfection degree as modules over a skew polynomial ring. We axlotnatlze the correspondltlg theory and ue show that i t is complete and that ~t adnilts quantifier e l ~ n i ~ n a t ~ o n In the usual nlodule language augmented with additwe functions whlch are the analog of the p-component functions. $

Nisan (STOC 1991) exhibited a polynomial which is computable by linear-size non-commutative circuits but requires exponential-size non-commutative algebraic branching programs. Nisan’s hard polynomial is in fact computable by linear-size “skew circuits.” Skew circuits are circuits where every multiplication gate has the property that all but one of its children is an input variable or a scalar....

A [ An−1 + p1A n−2 + · · ·+ pn−1 In ] = −pn In . Since A is nonsingular, pn = (−1)n det(A) 6= 0; thus the result follows. Newton’s Identity. Let λ1, λ2, . . . , λn be the roots of the polynomial K(λ) = λ + p1λ n−1 + p2λ n−2 + · · · · · ·+ pn−1λ+ pn. If sk = λ k 1 + λ k 2 + · · ·+ λn, then pk = − 1 k (sk + sk−1 p1 + sk−2 p2 + · · ·+ s2 pk−2p1 + s1 pk−1) . Proof. From K(λ) = (λ − λ1)(λ − λ2) . . ...