When \(A=3\), the positive integral solutions of so-called Markoff equation $$\begin{aligned} M_A:x^2 + y^2 z^2 = Axyz \end{aligned}$$can be generated from single solution (1, 1, 1) by action certain automorphisms hypersurface. Since Markoff’s proof this fact, several authors have showed that structure \(M_A(R)\), when R is \({\mathbb Z}[i]\) or orders in number fields, behave a similar fashion...

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

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively , in the context of (locally) essential domains.

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively, in the context of (locally) essential domains.

We prove a Łojasiewicz type inequality for a system of polynomial equations with coefficients in the ring of formal power series in two variables. This result is an effective version of the Strong Artin Approximation Theorem. From this result we deduce a bound of Artin functions of isolated singularities.