What is the set {$e\in(R/ I)\times(R/J): e$ is idempotent}?
I am trying to solve part (d) of Chaper 11 problem 6.8 from Artin's Algebra textbook. I have already solved a,b, and c.
Let $I$ and $J$ be ideals of a ring $R$ such that $I+J=R$
(a) Prove that $IJ=I\cap J$
(b) Prove the Chinese Remainder Theorem: For any pair a,b of elements of $R$, there is an element x such that x is congruent to a (mod I) and x is congruent to b (mod J)
(c) Prove that if $IJ=0$, then $R$ is isomorphic to the product ring $(R/I)\times (R/J)$.
(d) Describe the idempotents corresponding to the product decomposition in (c).
Now, according to my interpretation, (d) is just asking us to compute the set {$e\in(R/ I)\times(R/J): e$ is idempotent}, which I will call $S$.
My attempt:
I tried to prove the claim $S=$ {$(1-i+I,i+J):i\in I$ and $1-i\in J$}. I was able to prove $\supseteq$. Then I attempted to show $\subseteq$ as follows: Suppose $(a+I,b+J)\in S$ where $a,b\in R$. Then $(a+I,b+J)(a+I,b+J)=(a+I,b+J)$. So $(a^2+I,b^2+J)=(a+I,b+J)$. So $a^2-a\in I$ and $b^2-b\in J$. But I do not know what to do next. In fact, I am not 100% sure if my claim is even true. Please help me!
Hint: first prove that, if $A$ and $B$ are any rings, the idempotent elements of $A \times B$ are the elements of the form $(x, y)$ where $x$ is an idempotent of $A$ and $y$ is an idempotent of $B$. Now apply this to the case when $A = R/I$ and $B = R/J$. (You will need to revise your claim.)