Show that N= (3x, y)|x, y ∈ Z  is a maximal ideal ofZ×Z. Note that you must show that it is indeed an ideal.

Q: Let R (xy + yz, x2 – 3z?) be an ideal in the ring K[x, y, z] is R is radical ideal

A: Given: I(X) = (xy+yz, x^2-3z^2). Clearly, xy+yz, x^2-3z^2 vanish on X. Conversely, if a polynomial…

Q: Question 10

A: The given ideal is I=a,0/a∈ℤ.To prove that:I is a prime ideal, but not a maximal ideal of the ring…

Q: In Z[r], let I = {f(x) E Z[r] | f(0) is even}. (a) Prove that I = (x, 2). (b) Show that I is maximal…

A: Given, In Zx, let I=fx∈Zx | f(0) is even (a) Prove that I=x, 2 Let…

Q: Show that I = Z × {0} × ZL = {(a,0, b) : a,b E Z} is a prime ideal of R = Z × Z × Z but it is not…

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Q: In R[x], prove ideals (x3 – 1, x7 – 1) = (x – 1, x – 1).

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Q: 17. Find a nontrivial proper ideal of Z x Z that is not prime.

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Q: Let S = {a + bi : a, b e Z, b is even}. Then S is %3D not an ideal of Z[i] an ideal of Z[i]

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Q: O In Z07, let I = {la,0) :acz?. Show thet I is a prime ideal but not a manimal ideal. %3D

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Q: The ring Zs(i) has no proper ideals True False

A: Given statement is false. Justification is given in step two.

Q: Prove that subset S = {a + bi ∈ Z[i] | b is even} is not an ideal

A: A ring R is a set with two binary operations, addition and multiplication, satisfying several…

Q: The ring Z,2, has exactly-- --maximal ideals O 2 3.

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Q: Let I = {(x, y) |1, y E 2Z}. (a) Show that I is an ideal of Z × 2Z. (b) Use FIT for rings to show (Z…

A: We will show both parts.

Q: The ideal I Ð J is a prime ideal of Z Ð Z if O A. I = J O B. I = 0 and J is prime in R2 O C. I = Z…

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Q: Let T = Z - 5Z. Show that ZT is a local ring. What is its unique maximal ideal?

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Q: やthe ring KLx,り,z3 wherc K is a field. Prove (x2 - (y1)) that is a K[x, り,そI . Prime ideal of

A: Given:- Ring  K[x ,y ,z]   where k is a field. To Prove:-  [x z-(y2+1)]

Q: Let A and В be ideals in principal ideal domain R , where A + R and В + R . Prove that the ideal AB…

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Q: It homemorphisen from is to S. Then Prove that ģ ( Implis an ideal oft R.

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Q: а. Let I = {a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of the…

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Q: 4. Prove that I[a] = {ao + a1x + ..+ ana" a; = 2k; for k; e Z}, the set of all polynomials with even…

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Q: Let I = {a + bi: a, b e Z[i]: 3 divides both a and b}. Prove that I is a maximal ideal of the ring…

A: Given that  I=a+ib|a,b∈ℤ and 3 divides both a and b To prove I is maximal ideal of ℤi. To prove this…

Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: The ring Zs[i] has no proper ideals

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Q: 2. Let S= {a + bi | a, b e Z, b is even}. Show that S is a subring of Z[i], but not an ideal of…

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Q: Prove that the ideal (x2 + 1) is prime in Z[x] but not maximal in Z[x].

A: To prove that the ideal <x2+1> is prime in Z[x] but not maximal in Z[x]. Use the definition,…

Q: Q2: Let R a, b, c E Z}and let p: R → Z be defined such that = a. 4. Show that ker () is a prime…

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Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?

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Q: show that Q[x]/(3x⁴+2x³+1) is a field. Here (3x⁴+2x³+1) is the principal ideal generated by a…

A: Given factor ring is ℚx3x4+2x3+1 where 3x4+2x3+1 is the principal ideal generated by the polynomial…

Q: Let be R an integral domain and let / be an ideal in R, then R/Iis also an integral domain. Select…

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Q: The ring Zs[i] has no proper ideals True O False O O

A: The given statement is ,The ring Z8i has no proper ideals.

Q: R/H is integral domain, if and only if H is.........of R. O (i) maximal ideal O (ii) prime ideal O…

A: Given:  Suppose that R is an integral domain. Then R has at least two elements (since 1≠0) and hence…

Q: - If A is an ideal of R and x, y e R then prove that (i) x e A A+x=A (ii) A +x = A +y e x-ye A.

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Q: i) A = { ): : a, b e Z is a left ideal of R, but not right ideal of R.

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Q: Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of…

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Q: The maximal ideal of ZXZ is a) Zx 2Z b) ZxZ c) Zx3Z d) 3ZxZ

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Q: UIF 4 is a Ting homomodphism. Som R Los Then that 9(kes d) is Lis fture jdeal ofs. Ci fdove that '…

A: Given ϕ:R→S is a ring homomorphism, where R, and S are rings.   Definition: Ideal : Let R be a…

Q: [a Prove that I= { b. : a,b,c,d are even integers} is an ideal of M2(Z). d_ 1.

A: 1) The main aim is to show that I=acbd|a,b,c,d are even integersis an ideal in M2Z. to show that the…

Q: Let D be a principal ideal domain and let p E D. Prove that (p) is a maximal ideal in D if and only…

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Q: Let (a) Show that I is an ideal of Z × 2Z. (b) Use FIT for rings to show (Z × 2Z)/I ≈ Z₂. I= {(x, y)…

A: Given: Let us consider the given I=x,yx,y∈2ℤ a) Let us depict an ideal of ℤ×2ℤ b) Let us depict…

Q: In Z{x}, Let I = {f(x) ∈ Z[x] / f(0) is an even integer} Prove that I = is I prime ideal of Z{x}?…

A: It is given that,  ℤx is a ring and I = f(x)∈ℤx : f(0) is an even integer. (i) Prove that I = x, 2…

Q: If U, V are ideals of a ring R, let U + V = {u+ v:u E U,v E V}. Prove that U +V is also an ideal.

A: We have to prove the conditions of ideal

Q: Recall that the Gaussian integers Z[i] is the following subring of C: Z[i] = {a+bi a,b ≤ Z}, where…

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Q: Use the Cauchy Residue Theorem to find ∫C [3(z+1)] / [z(z-1)(z-3)] for C being z(t) = 2eit

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Q: The ring Zpq?r has exactly------------maximal ideals O 3 2

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show that the ideal A = {xf (x) + 2g (x) : f (x), g (x) e Z [x]} %3D is a maximal ideal of Z [x]

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Q: Let R = Z3[x]and let I = {q(x)(x² + 1)|q(x) E Z3[x]} then I is O Maximal ideal of R O Not prime…

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Q: Let I and J be ideals of R. Prove that if I ∩ J = {0}, then rs = 0 for r e I and s e J.

A: Given- I and J are ideals of R. To prove- If I∩J={0} then rs=0 for r ∈ I and s ∈ J

Q: Prove that I= (2+ 2i) is not a prime ideal of Z[1].

A: It is given that, I=2+2i is not a prime ideal of ℤi. Definition used: A prime ideal is an ideal I…

Q: b] { c d [a 1. Prove that I= a,b,c,d are even integers} is an ideal of M2(Z). %3D

A: