I have a problem which ask me to verify that to structures are rings. However, I'm unsure of how exactly to check each property. I believe that the first is closed but not sure how to check the additive identity, Inverse, associtivity on ⊗ and distribution of ⊗ over ⊕. For the second one I have no clue because I think it is not closed. Pleased offer some assistance.
(1) $S=Z$; for all $a, b∈S$,$ a⊕b=a+b+1$ and $a⊗b=ab$
(2) $S ={k/2|k∈Z}$={$...,−5/2,−2,−3/2,−1,−1/2,0,1/2,1,3/2,2,5/2$}; for all $a,b∈S, a⊕b= a+b$ and $a⊗b=2ab$.
(1) is not a ring since $$a \otimes (b \oplus c) = a(b+c+1)=ab+ac+a$$ while $$(a \otimes b) \oplus (a \otimes c) = ab+ac+1$$ so equality does not hold for $a \neq 1$.
(2) is a ring because $(S, \oplus)=(\mathbb{Z}/2, +)$ is an abelian group, $\otimes$ is clearly associative since $$a \otimes (b \otimes c) = 2a(2bc) = 4abc = 2(2ab)c = (a \otimes b) \otimes c$$ and distributivity holds since $$a \otimes (b \oplus c) = 2a(b+c)=2ab+2ac=(a \otimes b) \oplus (a \otimes c)$$