Let $R$ be a commutative semiprime ring with $1_R$. Suppose that $R$ satisfies the follwoing property: For each $a\in R$, there exist $b,c \in R$ such that $l_R(a) = Rb$ and $Ra = l_R(c)$, where $l_R(x)=\{r\in R:ra=0\}$. Rings satisfying this property include: Regular rings, that is, if $aba = a$ then $l_R(a) = l_R(ab) = R(1 - ab)$ and $Ra = Rba = l_(1 - ba)$, with a similar computation on the right. Also/and uniserial rings of finite length.
My question is the following:
(1) If $R$ satisfies the property above, does every ideal $I\subseteq R$ satisfy the same property?
(2) Does the quotient ring $R/I$ satisfy the property too?