Solving Rings and Idempotent Problems - RK

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I have a tablet so I have made a PDF of all my work and the problem. the file is attached to this post. please let me know if i am on the right track or give me a hint. I am currently stuck in attempt 2 and don't like my solution in attempt 1.

attempt 1: at the very last step I am using multiplicative inverses and I haven't proved that they must exist. but since I have shown that a multiplicative identity is required, it shouldn't be hard to prove that inverses exist also but i don't know if it will be required..?

attempt 2: took a different approach at the problem, and while it's a bit more complicated in the end and is unfinished (this is where i am stuck), I think it's the better attempt.



thank you,
RK
 

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a^3 = a implies that aaa = a.
Multiply by "a inverse" to obtain aa=1.
Multiply by "a inverse" again to obtain a=a^-1.
So each element is it's own inverse.
 

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