February 4th, 2018, 11:23 PM  #1 
Member Joined: Sep 2011 Posts: 83 Thanks: 1  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. 
February 4th, 2018, 11:50 PM  #2 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
You shown in (a) that $$a \oplus 0 = a1$$ 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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