6. Indicate whether each of the following statements is True (T), or False (F). Explain your answers. a) The ideal generated by a subset S subset and equal R is the largest ideal of R that contains S...... b) There exists an integral domain with characteristic 4...... c) Every ring with unity has at least two units.......

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: 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: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is a…

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

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Q: Prove that the ideal <x2 + 2> is prime in Z[x] but not maximal in Z[x].

A: The prime ideal <x2+2> is not maximal in Z[x] if Z[x]/<x2+2> is isomorphic to the…

Q: Let i, j be coprime positive integers, and let R be an integral domain. Prove that (r-y) is a prime…

A: To show that the ideal generated by x^i-y^j under the given conditions is a prime ideal in the ring…

Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then... ..... .... (a) 1 = R (b)…

A: {R, +, .) is a ring and it is given that I is not a maximal ideal in R.From this, we conclude that I…

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: ) Let Z3[i] = {a+ bi | a, b E Z3}. Show that Za[i] is isomorphic to Zs[a]/ (a² + 1). ) Is the ideal…

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

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Q: Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only…

A:

Q: (c) Show that the ideal generated by x² + y² + z² € C[x, y, z] is a prime ideal.

A: If <x2+y2+z2>is not a prime ideal, there should exist g,h∈ℝx, y, z s.t. x2+y2+z2 | gh,…

Q: Abstract Algebra. Please explain everything in detail.

A: To describe all the field automorphisms of the given field.

Q: 10- If F is a field then every ideal of FTx] is principal ideal domain. a) True b) False O a) True O…

A:

Q: 1. Consider the ring Z[r]. Prove that the ideal (2, x) = {2f(x)+rg(x) : f(x), g(x) E Z[r]} is not a…

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

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

A:

Q: 9.16. Let R be a ring and I a proper ideal. 1. If R is an integral domain, does it follow that R/I…

A: Let R be a ring and I be an ideal. 1. Choose R=(ℤ, +, ·) and I=4ℤ. Result: Ideals of ℤ are nℤ where…

Q: 16. Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z*; n2 1} Show…

A:

Q: Establish the following assertion there by completing the proof of Theorem 3-28: If (F , +,.) is a…

A: (F,+,.) is a field of characteristic zero and (K,+,.) the prime subfield generated by the identity…

Q: For every nE in the (Z , + , . ) ring, the I=nZ subring is an ideal. please show

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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: 5. Suppose that (R, +,.) is an infinite commutative ring and it has no nontrivial ideals, then R…

A: 5.R,+,.is an infinite commutative ring and it has no nontrivial ideals then R forms 6.suppose that…

Q: 1.29. Let f = x² + x + 1. (a) Is the ring F7[x]/(f) an integral domain? (b) Show that Z[x]/(7) =…

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Q: The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3

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Q: Prove that the ideal <x> in Q[x] is maximal.

A:

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

A: Yes it is true .

Q: Show that for any nontrivial ideal I of Z[i], Z[i]/I is finite.

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

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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…

A:

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

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Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J…

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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: 1. Suppose that (R, +,.) is a ring and I is not maximal ideal in R. Then .. (a) I = R (b) 3/ ideal…

A:

Q: The ring Z8[i] has no proper ideals True False

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

Q: 4Z is prime ideal of 2Z, but it is not Maximal ideal of Z. To fo

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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: Prove that Z[x] is not a principal ideal domain.

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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: Consider the ring of polynomia! Q [ Z] 2 -1 €Q[x] is aprincipl ideal 5

A: In the question it is asked to find whether <x2-1> is a principal ideal.

Q: 1. Let D be a quasilocal PID (that is, a PID with a unique maximal ideal). Prove that for all a, b e…

A:

Q: If I1 and I2  are two ideals of the ring R, prove that Ii n 11 ∩ I 2  is an ideal of R.

A: Given  I1 and I2  are two ideals of the ring R To prove : I1∩I2 is an ideal of R.

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: 12. Show that the set of all constant polynomials in Z[r] is a subring but not an ideal of Z[x]

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Q: If F is a field, then it has no proper ideal. T OF

A: I have given the answer in the next step. Hope you understand that

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: Show that the ideal I = (6) is a maximal ideal of E.

A: Show that the ideal I = (6) is a maximal ideal of E.