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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.

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"Do you have any idea or any source for finding a prove?" Yes. You could try assigning this as a homework problem. Any competent students should certainly be able to provide a proof for you. –  Steven Landsburg Dec 14 '10 at 19:34
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this question already appears verbatim on math.stackexchange, together with a partial proof. –  Peter McNamara Dec 14 '10 at 19:42
    
Peter beat me to it. Here's the link: math.stackexchange.com/questions/14313/ring-theory-lift I'm voting to close. Clearly homework. –  José Figueroa-O'Farrill Dec 14 '10 at 19:46
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It is worth noting that the original poster has posted several questions of the same ilk, and been told each time that the question is not appropriate for MO: see mathoverflow.net/faq#whatnot –  Yemon Choi Dec 14 '10 at 20:55
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closed as no longer relevant by Andreas Thom, José Figueroa-O'Farrill, S. Carnahan Dec 14 '10 at 19:54

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