How do regular and unit regular elements interact with the direct sum.

Question

Suppose $R$ is some (possibly noncommutative) ring. I say that an element $a\in R$ is regular if there is some $x\in R$ such that $a=axa$. I say it is unit regular if I also require $x\in U(R)$, the units of $R$. Suppose $S$ is some other ring, how do the regular and unit regular elements interact with the direct sum operation? Suppose we know what the regular and unit regular elements of $R,\,S$ are; can we say anything about the regular and unit regular elements of $R\oplus S$? Suppose all the regular elements of $R,\,S$ are unit regular, is the same true for $R\oplus S$? My guess is no, but are there certain conditions upon which this is true? I was thinking of applying this to the case where $R=\mathbf{Z}[G]$, $G$ finite, and $S=\mathbf{Z}$.