Choose a finite set of more than 2 elements and having at leastone prime less than 2021. Call such set S. Then define two suitableoperations (nontrivial) on the power set P(S) of S, call themaddition and multiplication. Then:

(i) show that P(S) is a ring under these operations.

(ii) 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 agroup under multiplication.

(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?

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