Структурные графы колец: определения и первые результаты

Abstract

The quadratic Vieta formulas (x,y)↦(u,v)=(x+y,xy) in the complex geometry define a two-fold branched covering C2→C2 ramified over the parabola u2=4v. Thinking about topics considered in Arnold's paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of the complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ:A2→A2 defined by the Vieta formulas (x,y)↦(u,v)=(x+y,xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate simplest finite rings A=Zm and A=Zk×Zm. Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.