One of the graphs associated with any ring R is its distant graph G(R, Δ) with points of the projective line P(R) over R as vertices. We prove that the distant graph of any commutative, Artinian ring is a Cayley graph. The main result is the fact that G(Z, Δ) is a Cayley graph of a nonartinian commutative ring. We indicate two non-isomorphic subgroups of P SL2(Z) corresponding to this graph.Mathematics Subject Classification. Primary 05C25, Secondary 51C05.

There is presented an infinite class of subgroups of the modular group PSL(2, Z) that serve as Cayley representations of the distant graph of the projective line of integers. They are infinite countable free products of subgroups of PSL(2, Z) isomorphic with Z2, Z3 and Z subject to the restriction that the number of copies of Z is 0 or 2. The proof technique is based on a 1-1 correspondence between some involutions ι of Z that fulfill the equation ι(ι(n) − δn) = ι(n + 1) + δn+1, δn = ± 1, δ ι(n) = δn, and groups from this class.

Abstract. Given the algebra T of ternions (upper triangular 2 × 2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T 2 . This set of points can be represented as a set of planes in the projective space over F 6 . We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F -linear antiautomorphism, the plane model of our projective line does not admit any duality.

Abstract. We discuss the free cyclic submodules over an associative ring R with unity. Special attention is paid to those which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular 3 × 3 matrices over a field F under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.Mathematics Subject Classification. 51B99, 51C99, 51E25.

Abstract. We classify symmetric 2-structures