9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 — х€R" (c) (1+x) # R (d) No Choice (a) (b) (c) (d)

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

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Q: 3. Let R. be a commutative ring with ideals L and M. Give an example of when LMLn M.

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Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

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: If r is commutative ring then deg(f(x) + g(x)) = max(deg(f(x)),deg(g(x). False True

A: False

Q: Consider (Z. +, O) where + is usual addition, O is define a O b=6. Then (Z, +. O) is not ring…

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Q: 7. Let R be a commutative ring and let F = F(R, R) be the set of functions f: R→ R. Functions in F…

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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: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab = ba. Then (a) (a +…

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Q: If R is commutative ring then deg((x) + g(x)) = max(deg(f(x)),deg(g(x)). %3D False True

A: We know that deg (f(x) + g(x)) is less than equal to max (deg(f(x)), deg(g(x))) .

Q: If R and S are rings, define the direct sum of R and S, ROS by RS = {(r, s) : r € R, s € S} where…

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Q: 5) Suppose that (R, +,.) be a ring without identity and has a subring with identity, then (a) R…

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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.…

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Q: 4. Write out the addition and multiplication tables for each congruence class ring below, then…

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Q: Let r, s, and t belong to a ring R. Then which one of the following is correct? (a) r0 = 0 (b) -(rs)…

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Q: 14. Consider the quotient ring R = GF₁[x] / where me GF₁. For any specific value of m, the number of…

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Q: 3. In a Ring (R, +, *) let '0' be identity of '+' and '1' be identity of only one element. unique…

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Q: Prove that in a Euclidean ring R, (a, b) can be found as follows : b = q0 a + r¡, where d (r¡)<d (a)…

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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…

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Q: Which one of the following is true? In the ring Z6, 4 does not divide 2 In the ring Z6, if x² = x…

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Q: 2. Determine whether or not the set is an ideal in the ring M2×2(R). J-{(8₂9) | PCR} = ER

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Q: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab = ba. Then (a) (a+…

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Q: 1. If (R, +,.) be a ring without divisors of zero then: (a) R has 1 (b) the cancellation law holds…

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Q: If R is commutative ring then deg(F)g(x)) = deg(f(x)) + deg(g(x)). False O True

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Q: If R is commutative ring then deg((f(x) + g(x)) = max(deg(f(x)), deg(g(x)). O False O True

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