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February 4th, 2018, 10:23 PM   #1
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Rings Problem

Hello all, I have done part (a) of the question as attached and am not sure if they are correct. Would appreciate if you can help me to see. Next, I have no idea how I should do part (b). Greatly appreciate! Thanks in advance!

Axiom 4 and 5 are found in the written image.
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February 4th, 2018, 10:50 PM   #2
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You shown in (a) that
$$a \oplus 0 = a-1$$
this is NOT what you want. Hence $0$ is NOT the additive identity.

You need to find some element $e$ (not necessarily equal to $0$) such that
$$a \oplus e = a~\text{and}~e\oplus a = a$$
for ALL $a$ in the ring. So you need to actually find such an $e$.
And in the second part of (a), you actually need to find such an $a^{-1}$.

Your proof should start with: define $e=....$, then we show that this works. And also for the second part of (a).
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