Unit-free commuting PI- regular Rings

Let R be an associative ring with Unitary and U denoted the set of all invertible element of R. we say that if for every x, y ∈ R\U there exist a positive integer n and a∈R such that (xy) n= (yx) n a (yx) n.

then R is a Unit-free commuting π-regular ring. we show that if R is a Unit-free commuting π-regular ring, then for any e2=e, eRe is Unit-free commuting π-regular.

In this paper shown that the center c(R) of every Unit-free commuting π-regular ring

R is again Unit-free commuting π-regular.

We also proved that evey Unit-free commutingπ-regular ring, then R is π-regular.