The connected components on the projective line over a ring (Q2726404)

Let \(R\) be a ring. A point of the projective line over \(R\) is a module of the form \(Ra\) where \(a\in \mathbb{R}^2\) admits some \(b\) such that \((a,b)\) is a basis for the free \(R\)-module \(\mathbb{R}^2\). Points \(A,B\) of the projective line over \(R\) are called distant if \(A=Ra\) and \(B=Rb\) for a basis \((a,b)\) of \(\mathbb{R}^2\). Points \(A,B\) are called connected if there is a chain \(A=A_0\), \(A_1, \dots,A_n=B\) of points such that \(A_i\) distant \(A_{i+1}\). NEWLINENEWLINENEWLINEThe authors prove that any two points are connected if and only if \(R\) is a \(GE_2\)-ring. If \(R\) is the ring of endomorphisms of an infinite-dimensional vector space it is proved that any two non-distant points \(A,B\) admit points \(C,D\) such that: [A distant \(C\) and \(C\) distant \(B]\) or \([A\) distant \(C\) and \(C\) distant \(D\) and \(D\) distant \(B]\), and the first case does not apply for all points \(A,B\).