R is called commuting regular ring (resp. semigroup) if for each x,y $in$ R there exists a $in$ R such that xy = yxayx. In this paper, we introduce the concept of commuting $pi$-regular rings (resp. semigroups) and study various properties of them.

The relation $$xy-yx=h(y)$$ , where $$h$$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach rigorous meaning to right-hand side this equality, we assume that $$x$$ and $$y$$ are elements Banach algebra (or an Arens–Michael algebra). We prove universal generated by commutation kind can be represented explicitly as analytic Ore extension. An analysis...

we study topological von neumann regularity and principal von neumann regularity of banach algebras. our main objective is comparing these two types of banach algebras and some other known banach algebras with one another. in particular, we show that the class of topologically von neumann regular banach algebras contains all $c^*$-algebras, group algebras of compact abelian groups and cer...