1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx € / for every x,y E R. Deduce that if K, and K₂ are ideals of R and both R/K, and R/K₂ are commutative, then R/(K₁ K₂) is also commutative. 2. Suppose that D is an integral domain and that J and K are ideals of D neither of which equals {0}. Show that Jn K = {0}.

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Q: 9. Let R be the ring Q[x]/(x³ - x² − x + 1). (a) List all the ideals of R and draw a diagram showing…

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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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Q: 16.2.12. Let I be an ideal in a commutative ring R. Prove that I[x] is an ideal in R[x]. Prove that…

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Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

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Q: Let R be a commutative ring with unit which is not an integral domain. Show that quotient and…

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Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

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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: For any ideal (1, 1.) of the ring (R, 1.), lefine the set V by Vi - {rER " E I for some nEZ4}. %3D…

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Q: 18.6.1. Let R be a non-trivial commutative ring with identity. Show that the following are…

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Q: 8. If 1 and 2 are two ideals of the ring R, prove that the set 1₁+= {x+yx Eh, yЄh} is an ideal of R…

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Q: 4. Let p: R S be a ring homomorphism. Show that J = ker p is a prime ideal if S is a domain. Show…

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Q: . Consider the ring Z with ideal . Give the ring tables of Z/ .

A: Consider the ring  with ideal HereLet R = We know thatAndWe have to write the addition and…

Q: Let R be a ring and consider RxZ={(r,n)|r e R, n e Z}. Define (r,n) + (s,m) = (rts, n+m) (r,n)(s,m)…

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Q: 6. Find two ideals I₁ and 12 of the ring Z such that a. I, U2 is not an ideal of Z. b. I₁ U2 is an…

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

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Q: 1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т…

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Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

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Q: 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.…

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Q: 2. Let R[x] be a ring over field R and let f, g are elements of R[x]. f=x3 +x2 +x +[0] , g=x +[1].…

A: We are given : f(x)=x3+x2+x+0⇒f(x)=x3+x2+xand g(x)=x+[1]⇒g(x)=x+1Now, Dividing f(x) by  g(x), we…

Q: IN denotes the set of noninvertible ele conditions are equivalent: (N,+,) is an ideal of (R,+, ), p)…

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Q: 5. Let F be a field and 0: F – R be a ring epimorphism. If Kere F, show that R has no zero divisors.

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Q: 1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx J…

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Q: = d With A = 29, B= 20, U = A.B 580, let J be (-290.. 290]. There is a ring-iso g:ZAXZB → Zu sending…

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Q: 11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) +…

A: Let x, y ∈ R. Then (fg)(x + y) = f(g(x + y)) = f(g(x) + g(y)) = f(g(x)) + f(g(y)) =(fg)(x)+(fg)(y),…

Q: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…

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Q: Let an ideal I = {(a, 0): a ∈ Z} of the ring R = Z ⊕ Z be given. I, R is a prime Is it the ideal

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