Assign a grade of A (correct), C (partially correct), or F (failure). Justifyassignments of grades other than A.(a) Claim. If is a ring, then the equationhas a unique solution.Proof. Suppose p and q are two solutions to Thenand Therefore, Therefore, (b) Claim. If is a finite integral domain, then is a field.Proof. Suppose R has n elements. Let Then the powersof are not all distinct. Therefore, forintegers t, r, where we may assume that Then andtherefore, Thus, Therefore, x has an inverse.Hence, R is a field.

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