Is every ideal of a unit regular ring unit regular?

Question

A ring $R$ with unity $1_R$ is unit regular if for any $a\in R$, $a=aua$ for some unit element $u\in R$. An ideal $I$ of a ring $R$ is a unit regular ideal if for any $x\in I$, there exists $u\in R$ such that $x=xux$.

My probem: Is every ideal of a unit regular ring unit regular? If not, are there examples of regular ideals of a unit regular ring that are not unit regular?

Answer 1

From what you've written, it's trivially the case that:

1. an ideal contained in a unit-regular ideal is unit-regular; and
2. the ordinary definition of "unit-regular ring" is just that $R$ is a unit-regular ideal of $R$; therefore
3. Every ideal of a unit-regular ring is a unit-regular ideal.