Define a ring R to be the integers with the operations ab=a+b-1 ab:= a + b - ab, where the operations on the right are the usual operations on Z. Prove that Ris a ring.

Demonstrate or explain why the system (P(R#), +, *) is NOT a field, that is, demonstrate or explain why some elements in set P(R#) do not have inverses corresponding to the * operation.

(b) Does [68]183 have a multiplicative inverse in the ring Z183? Find it if so, or explain why if not.

Define a new addition ⊕ and multiplication ⊙ on ℚ by               a ⊕ b = a + b + 1 and a ⊙ b = ab + a + b, where the operations on the right-hand side of the equal signs are ordinary addition, subtraction, and multiplication. Prove that, with the new operations, ℚ is a commutative ring with identity. Is it an integral domain.Provide a detailed and well explained proof