Help with Part (a) & (b) of Question - Appreciate Assistance!

[Joe20]

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!

 

[Euge]

Hi Alexis87,

Your argument for part (a) is incorrect. You assumed $O_R = 0$, but it's $1$. First note that $\oplus$ is commutative. Now given $a\in R$, $a\oplus 1 = a + 1 - 1 = a$, and hence $1 = O_R$. The additive inverse of an element $b\in R$ is $2 - b$ since $(2 - b) \oplus b = (2 - b) + b - 1 = 1$. This shows axioms 4 and 5 are satisfied.

To answer part (b), show that $a\odot b = 0_R$ in $R$ implies $a = 0_R$ or $b = 0_R$, i.e., $a\odot b = 1$ implies $a = 1$ or $b = 1$. The equation $a\odot b = 1$ is equivalent to $ab - (a + b) - 2 = 1$, which is equivalent to $ab - (a + b) - 1 = 0$. Factor the left-hand side of the latter equation to deduce $a = 1$ or $b = 1$.