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.

Q: Show that the set Q(√2) = {a+b√2, a, b € Q} is a ring, under operations of multiplication and…

A: The set Q2=a+b2 , a , b∈Q  and operations under multiplication and addition define by :…

Q: Let R = {0, 2, 4, 6, 8}. Then R is a ring under addition and multiplication (mod 10). Find the…

A:

Q: Let us define two new operations on R called ~ and ∗ by the following formulas. For real numbers a…

A:

Q: List the elements of the quotient ring Z2 [x]/(x² + 1) and make an addition and multiplication table…

A:

Q: Theorem 79. Let a, b, c, d ∈ R where R is a ring then (a + b)(c + d) = ac + ad + bc + bd. Note.…

A:

Q: Let R be a commutative ring with identity, and let a, b ∈ R. If a² + b² = 0, prove that (a + b)² =…

A: In algebra, a ring is a set equipped with two binary operations - addition and multiplication, that…

Q: Can someone please help me understand the following problem. I need to know how to start the…

A:

Q: 3. Consider the set {a+b√2: a, b E Z} with the usual addition and multiplication. Show that this is…

A:

Q: J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1…

A:

Q: Enter the multiplicative inverse of the element [21]49 X+ [18]49 in the ring Z49[x].

A: Extended Euclidean Algorithm

Q: Let R be a ring with unity. Assume b1a=1R and ab2=1R. Prove that b1=b2.   Also, let R be a ring…

A: Ring with unity,so 1R is different from 0R ,

Q: X a b с a b a C

A:

Q: If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an…

A:

Q: Construct the multiplication table for the ring Z/5Z.

A:

Q: Define a ring R to be the integers with the operations ab=a+b-1 ab:= a + b - ab, where the…

A: The given operations,The given set  is the set of all integers.The aim is to prove that the set of…

Q: Question 2. (a) Prove that [65] 186 has a multiplicative inverse in the ring Z186. (b) Compute this…

A: (a) To Prove: 65 has a multiplicative inverse in ℤ186. (b) To Find: Inverse of 65 in ℤ186.

Q: Let R = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

A:

Q: Define a new addition ⊕ and multiplication ⊙ on ℚ by               a ⊕ b = a + b + 1 and a ⊙ b = ab…

A:

Q: Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 =…

A: Given Suppose that R is a ring with nonzero-divisors and that R contains a nonzero element b such…