If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an example to show that this may fail if the ring is not commutative( maybe using a counterexample (a+b)6=0)

Let P be a prime ideal and R a commutative ring with identity. If R/P is an integral domain, explain why R/P cannot be the ring with one element. Also explain why P is not equal to R in this scenario.

Enter the multiplicative inverse of the element [22]44 X+ [25]44 in the ring Z44[x]. When you enter your answer, make sure that all congruence classes are in the form [a]44 with 0 ≤ a<44. You don't have to type the brackets. For example, if your answer is [2]44 x² + [3]44, you can type 2x^2+3. Answer:

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