Unit-free strongly commuting Regular Rings

In this paper, ring R satisfying in the condition xy = (yx)2a(yx)2 for all x; y 2 R n U with some a in R and is called Unit-free strongly commuting Regular Rings. We observe the structure of a Unit-free strongly commuting regular ring. In this paper shown that R is a Unit-free strongly commuting regular ring, then R is an abelian ring. we also proved that R is a Unit-free strongly commuting regular ring, then J(R) _ N (R) and shown that R is a local ring with J(R)2 = 0 and also, we proved some main properties of the Unit-free strongly commuting regular rings and we give a necessary and su cient condition that a ring is Unit-free strongly commuting regular. Keywords: regular rings; Strongly commuting regular rings; Unit free commuting regular rings; reduced rings