subject name (rind and fields)

promis) Choose a finite set of more than 2 elements and having at least one prime less than 2021. Call such set S. Then define two suitable operations (nontrivial) on the power set P(S) of S, call them addition and multiplication. Then: (i) show that P(S) is a ring under these operations. (ü) Also write Cayley's table for the both operations. (iii) Is the ring P(S) commutative? (iv) What is the unity if P(S) has unity? (v) Find units of P(S) and show that the set of units forms a group under multipli- cation. (vi) Are there any zero divisors in P(S)? (vii) Is there any idempotent? (viii) Describe principal ideals in P(S). (ix) Is P(S) a field?