Let $R$ be a ring, $A$ is an ideal of $R$, and assume that $A\subseteq J(R)$. Let $r\longmapsto \overline{r}$ denote the coset map $R\rightarrow \overline{R}=R/A$. Show that the following are equivalent:
(1) $A$ is lifting.
(2) If $\overline{R}=\overline{K_1}\oplus...\oplus \overline{K_n}$ where the $\overline{K_i}$ are left ideals of $\overline{R}$, there exists a decomposable $R=L_1\oplus...\oplus L_n$ where the $L_i$ are left ideals of $R$ such that $\overline{L_i}=\overline{K_i}$ for each $i=1,2,...,n$.
(3) Same as $(2)$ except $n=2$.\
Do you have any idea or any source for finding a prove?
**If $A$ is an ideal of $R$, and if $r+A$ is an idempotent in $R/A$, we say that $r+A$ can be lifted to $R$ if there exists an idempotent $e^2=e \in R$ such that $e+A=r+A$, that is if $e-r \in A$. We say that idempotents can be lifted moduleo $A$, or that $A$ is lifting, if every idempotent in $R/A$ can be lifted.
