Check if the solution is correct. Correct the mistakes. If x4=x for any x in R, prove that R is commutative

Q: Find the inverse of X + (X² – 8X + 7) in the form aX + b in the ring Q[X]/(X² – 8X + 7). - a b =…

A:

Q: The characteristics of local rings is either 0 or prime.

A: Let R be a local ring (with 1). Suppose char R = n ≠0. Let, if possible, n be divisible by two…

Q: 5. A division ring is a ring with identity in which every nonzero element has a multi- plicative…

A:

Q: Exercises :- 1- Show that: (Z24 / {0,6, 12 ,18},+,.) = (Z6, +6 r6) 2- Show that: (Z20 /{0,10},+,.)…

A: "Since you have posted a question with multiple sub-parts, we'll solve first 3 parts for you. To get…

Q: Q2: (A) Choose the correct answer for each of the following: 1. The cancellation law holds in a. any…

A: Answer 1: Must know:The cancellation laws hold in a ring R if an only if R has no zero divisors

Q: * (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no…

A: True

Q: Decide whether z(sqrt(-5)) unique factorization domain or not (ring theory)

A: To decide whether Z(√-5) is unique factorization domain or not.

Q: 29 justifying each step you use for this. 57 b) Calculate

A: (b) The objective is to find the value for 2957. By using the division algorithm, 29=0×57+29. So,…

Q: (b) Let R be a ring and S be a subring of R and show that OR = 0s, where OR, Os are the zero…

A: To Prove: Zero element of ring and subring is same.

Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

A:

Q: (a) Let R be a ring and OR E R be the zero element of R. Then check as to whether or not {OR} is a…

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Q: expr lim x→-8 -2x² - 20x - 32 x + 8 - 32)

A: Given problem:-

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

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Q: In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.

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

A: Extended Euclidean Algorithm

Q: Determine the splitting field of x4 + x2 + 1 over Q also find its degree over Q, Do not copy paste…

A:

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

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Q: Q 10 Which one of the following is not a ring? (a) (Z, +, ·) (b) (M4x4(R), +, ) (c) (Zs,O5, 85) (d)…

A: We know that of G is group then G is follow closure property .  G is follow associative property.…

Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: For any prime number p and any positive integer n, prove that every zero in the ring Z-n is…

A: We need to show that the every zero element in the ring is nilpotent element.

Q: Given a ring R with elements {0, 1, a, b}, where: a+a=b, a×a=b, and b+b=0. Find the result of…

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

Q: Q2: (A) Choose the correct answer for each of the following: 1. The cancellation law holds in a. any…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Check if the following subsets of real numbers are Rings or not. • {a + b*(2)^(1/3) | a, b ∈ Q} • {a…

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Q: Please use method from Abstract Algebra, Show that the polynomial 37a + 30a + a4 = 1500 does not…

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Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

Q: Let R be the set of all functions. It is known that (R,+,×) is a ring where + and × are both…

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Q: Write the multiplication table for the following commutative quotient ring: Z2[x]/(x³ + x² +x + 1).

A: Given that, ℤ2x/x3+x2+x+1 is a commutative quotient ring. We have to frame the multiplication table…

Q: Describe all ideals of the following quotient ring: Q[x]/(x² – 5x + 6)

A: Given:Q[x](x2-5x+6)