Let me add some detail to Pace's sketch.<p>

Let's say that a pair $(A,B)$ of ideals of $R$
*satisfies the conditions* if<p>

(i) $A$ and $B$ are proper,<br>
(ii)  $A+B=R$,<br>
(iii) there is no idempotent $e$ in $A$ whose complementary idempotent
$1-e$ lies in $B$. <p>

[Before beginning, note that Condition (i) follows from
Condition (iii): using Condition (iii) and the fact that
both $A$ and $B$ contain $0$, derive that neither
$A$ nor $B$ can contain $1$.]

**Lemma.**
If $(A,B)$ satisfies the conditions and $e\in A$
is idempotent, then $(A,B+Re)$ satisfies the conditions.<p>

**Pf.**
We establish Condition (iii) for $(A,B+Re)$
by reducing it to Condition (iii) for $(A,B)$.
Assume that $f$ is an idempotent
in $A$ whose complementary idempotent $1-f$
belongs to $B+Re$. Multiplying the expression
``$(1-f)\in B+Re$''
by $1-e$ yields that 
$(1-e)(1-f)\in B(1-e)+Re(1-e)=B(1-e)\subseteq B$,
so the idempotent $(1-e)(1-f)$ belongs to $B$.
But now, since $e, f\in A$, we have that the idempotent
$1-(1-e)(1-f)=e+f-ef\in A$, while its complement
lies in $B$. This contradicts Condition (iii) for $(A,B)$.

Condition (ii) for $(A,B+Re)$ holds because $B+Re$
extends $B$. \\\\\\

<p>

**Lemma.**
If $A$ and $B$ contain the same idempotents,
$(A,B)$ satisfies the conditions, and $e$
is an idempotent with $e, 1-e\notin A$,
then $(A+Re,B+Re)$ satisfies the conditions.<p>

**Pf.**
Let me argue only Condition (iii),
since (i) is a consequence of (iii) and (ii)
is obvious here.

Assume that there is some idempotent $f\in A+Re$
whose complementary idempotent $1-f$ lies in $B+Re$.
Multiply the expressions 
$f\in A+Re$
and 
$(1-f)\in B+Re$
by $1-e$. We get
$(1-e)f\in A(1-e)+Re(1-e)=A(1-e)\subseteq A$,
and $(1-e)(1-f)\in B(1-e)\subseteq B$.
Now, since we have assumed that $A$ and $B$
contain the same idempotents, and the idempotent
$(1-e)f$ belongs to $A$, we must have $(1-e)f\in B$.
Since 
both $(1-e)f, (1-e)(1-f)$ belong to $B$, the sum
$(1-e)f+(1-e)(1-f)=1-e$ belongs to $B$, contrary
to the assumptions of the lemma 
($A$ and $B$ contain the same idempotents and $1-e\notin A$). \\\\\\