Structure graphs of rings: definitions and first results

A. T. Lipkovski

Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 3, pp. 181-190

Résumé

The quadratic Vieta formulas $(x,y)\mapsto(u,v)=(x+y,xy)$ in the complex geometry define a two-fold branched covering $\mathbb C^2\to\mathbb C^2$ ramified over the parabola $u^2=4v$. 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 $\Phi\colon A^2\to A^2$ defined by the Vieta formulas $(x,y)\mapsto(u,v)=(x+y,xy)$. What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate simplest finite rings $A=\mathbb Z_m$ and $A=\mathbb Z_k\times\mathbb Z_m$. 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 $\Phi$.