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

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: Let I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.

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Q: Let I be a central ideal of L (i.e. I is contained in the centre Z(L). Prove that if L/I is…

A: For a set S, which forms an abelian group under operation addition and a structure denoted by [ ,]…

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: Let I be a central ideal of L (i.e. I is contained in the centre Z(L)). Prove that if L/I is…

A: Given that I is a central ideal of L. That is I is contained in the center ZL. Assume that L/I is…

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 R be a commutative ring with 1 and I be a proper ideal of R. Prove that I is prime if and only…

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

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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: 4. Let I and J be ideals of R. Prove that ifINJ={0}, then rs = 0 when reI and s eJ. %3D

A: Given that R be a Ring snd I.J be two ideals of R .Let r∈I and S∈ISince , r be an ideal and S∈J≤R…

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: Prove that the fields Q(257) and Q(V257i) Here, i = V-1. isomorphic but not equal. are

A: Here , Q is a set of rational number.  So , here we will prove that the given fields are isomorphic.…

Q: Let R be an integral domain. Prove that {0R} is a prime ideal.  Let R be a ring and let p ∈ R be…

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Q: Let A and B be ideals. Prove that A + B = {a+ ß: a E A, ß E B} is an ideal.

A: Let (R, +, ∘) is a ring and A and B are ideals of R.

Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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Q: Let D be an integral domain. (i) If aD is maximal,then show that 'a' is irreducible (ii) If D is a…

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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: Let R = Z4[x]. Let I = {f(x)eR|f(1) = 0} be an ideal of R. Then I is O Maximal ideal of R O Not…

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Q: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

Q: In the ring ], let I = (x²_8) O Prove that the quo tient ring [a]/Ix not an integral domain.

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Q: Prove that Z[2√-3] is not a principal ideal domain

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Q: a, be R H I 's an ideal of R if and anly of a-beI. for then a+I b+I %3D

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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: - 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: Let f (x) be a cubic irreducible over Zp, where p is a prime. Provethat the splitting field of f (x)…

A: Please see the proof step by step and

Q: Let R be a commutative ring. Prove that the principal ideal generated by the element x E R[x] is a…

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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: Let I= {(a, 0)| a eZ}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z. Id if…

A: We knew that an ideal I of a ring R is said to be prime ideal if for a ,b ∈ R and ab ∈ I this imply…

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

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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 f (x) be a cubic irreducible over Zp, where p is a prime. Provethat the splitting field of f (x)…

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

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Q: 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 Z[i].…

A: Given- S=a+bi| a, b ∈ℤ , b is even .To show that S is a subring of ℤ[i] , but not an ideal of ℤ[i].…

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: Prove that every ideal in F[x], where F is a field, is a principal ideal

A: To show: Every ideal in F[x], where F is a field is a principle ideal

Q: (a) Consider the inner product on P2 defined by (p, q) = aobo + 2a¡b1 + 3azb2 for each polynomial…

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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: 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 be an integral domain and I be an ideal of R. Then prove or disprove that R/I ⁄is an integral…

A: given  let R be an integral domain and I be an ideal of R  prove or disprove  RI is an integral…