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

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

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Q: Need correct answer, Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

A: Solution:-

Q: Q13: Define and give an example for (integral domain ring, zero divisor).

A: In this question we have to give definitions and examples.

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

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

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Q: The ring 3z is isomorphic to the ring 5z O False O True

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Q: How many distinct cosets are there in the factor ring Z[i] / ? Explain your answer.

A: Given: Let's determine the number of distinct cosets of the factor ring ℤi3+i.

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: Please use method from Abstract Algebra, Show that the polynomial 37a² + 30a³ + a⁴ = 1500 does not…

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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…

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Q: Explain why the polynomial rings R[x] and C[x] are not isomorphic.

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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: 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…

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Q: The rings Z and 5Z are isomorphic.

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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: N(8+i)N(8+i) = N[(8+i)(8+i)] = -. and 652 = (72 + 4*)² = N(7+4i)N(7+ 4i) N[(7 + 4i)(7 + 4i)] - 4.…

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Q: Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition…

A: Answer in steps:We need to show that (R₁ × R₂, +, ·) satisfies all the axioms for a ring.i.Closure…