1. Show that x^2 - y^2 = (x – y)(x + y) for all x, y in a ring R if and only if R is commutative. %3D

Q: Modern Algebra Let S = {[_$$]:q₂bER} -b Show that : CS given by a ,bER CM₂ (R) a (a + bi) = is a…

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Q: Let x, y, and z be an element of a ring Prove that x(y-z) = xy-xz and (y-z)x=yx-zx

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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: If Ris a commutative ring, then (b) a | b and b|c⇒a | ci.e., this relation of divisibil it y in Ris…

A: Given that if  R is a commutative ring, then we have to show that  (i) a|b and b|c then a| c i.e.,…

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

A: False

Q: 3. Let R be a commutative ring. Let x, y E R such that their characteristics are p. Show that (x +…

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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…

A: let R be a commutative ring and F=FR,R be the set of function f:R→R defined as f+gx=fx+gxfgx=fxgx…

Q: (8) Show that for a field F, the set S of all all matrices of the form a b (88) 00 for a, b F is a…

A: An ideal is a special kind of subring of a ring R. If a∈I and r∈R implies that ra∈I then, the…

Q: Q1/ Let (Maxz (IR), +) be a ring; Is it Integral domain ?

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Enter the multiplicative inverse of the element [22]44 x + [17]44 in the ring ℤ44[x]. When you…

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Q: Enter the multiplicative inverse of the element [6]18 X+ [5]18 in the ring Z18[x]. When you enter…

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Q: 5. Explain why each of the polynomials below is irreducible in the indicated polynomial ring. Make…

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Q: Can you me please Give an example of a ring R and polynomials f (x) and g(x) in R[x] such that…

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Q: = (19) Let om : Z → Zm be the ring homomorphism defined by σm(a) = remainder of division of a by m.…

A: Ans-19)We are given in the question (19) Let om Z→ Zm be the ring homomorphism defined by om (a) = =…

Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: Let R be a ring such that for each a belong to R there exists x belong to R such that a^2x =a. Prove…

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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: 1. The product of 3x²_5x² + 3 and 2x + 5x – 4 in z,[x]/ is а) 2x b) 2x + 1 c) 2x + 2 d) 2x + 3

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Q: 1. Let R be the ring of all functions f: R → R, with operations of pointwise addition and…

A: Ideal:  Let  is a ring. A nonempty subset  of a ring  is call an ideal of , if following are true.…

Q: Let R be a commutative ring with unity of characteristic 4. Compute and simplify (a+b)^4 for a and b…

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Q: よn aring of (2,+) find In <97

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Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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Q: 1. (12%) Let R be the set of all 2 x 2 matrices of the form where x is an integer. Prove that R is a…

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Q: True or False? Let R be a ring and a, be R. Then a^2 - b^2 = (a-b (a+b) v a,b € R

A: Given: a,b∈R and R be a ring then a2-b2=a-ba+b ∀a,b∈R To check that the given statement is true or…

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: (3) Let R denote a commutative ring with a one. An element x of Ris termed nilpotent if and only 3n…

A: We now solve the Q3

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

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