We study invertible generating pairs of fundamental groups of graph manifolds, that is, pairs of elements (g, h) for which the map g → g −1 , h → h −1 extends to an automorphism. We show in particular that a graph manifold is of Heegaard genus 2 if and only if its fundamental group has an invertible generating pair.

In this article we obtain 2 generalizations of the well known Gleason-Kahane-Z̀elazko Theorem. We consider a unital Banach algebra A, and a continuous unital linear mapping φ of A into Mn(C) – the n × n matrices over C. The first generalization states that if φ sends invertible elements to invertible elements, then the kernel of φ is contained in a proper two sided closed ideal of finite codimen...

If a and b are elements of an algebra, then we show that ab is Drazin invertible if and only if ba is Drazin invertible. With this result we investigate products of bounded linear operators on Banach spaces. 1. Drazin inverses Throughout this section A is a real or complex algebra with identity e 6= 0. We denote the group of invertible elements of A by A. We call an element a ∈ A relatively reg...

A weak group element is introduced in a proper *-ring. Several equivalent conditions of elements are investigated. We prove that an pseudo core invertible if it both partial isometry and invertible. Reverse order law additive property the inverse presented. Finally, under certain assumption on a, a◯Wa∗=a∗a◯W presented by using normality part its group-EP decomposition.

We describe decompositions of the group of units of a ring and of its subgroups, induced by idempotents with certain properties. The results apply to several classes of rings, most notably to semiperfect rings.