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What I've tried so far:

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0.3
is commutative monoid with multiply operation. for example, 3x2=3x4=0 in this group. however 2 is unequalt to 4. so it's not cancellative.

0.4 Prove that, for each , the ring is an integral domain if and only if n is prime.

(1) only if
if the ring is an integral domain, 0 is a unique zero-divior. If n is composite number, for s.t d | n and d>1, its order is n/d for add operation like d+...+d (add n/d times) = d*(n/d)(by distribution law) = 0. but it's contradiction to assumption s.t is an integral domain. So n is prime number.

(2) if
Let n be a prime number. suppose that there exists non-zero elements r in s.t r is zero-divisor. ie., n divides r*s. however, since , r*s have no prime factor n. so n does not divide r*s. it's contradiction. conclusively, is integral domain.
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Testingg