2. In the ring Z of integers, consider the principal ideal I = (3) = {3k|k E Z}. Find Z/I. %3D %3D

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Q: 38. Prove that I = (2 + 2i) is not a prime ideal of Z[i]. How many elements are in Z[i]/I? What is…

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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: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

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…

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Q: (3) The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +,.) (b) ((8), +,.) (c) ((12),…

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

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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: (7) In the ring (Z, +, .), we get n {P:P non trival prime ideal in Z}: (a) o (b) (Z, +, .) (c)…

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Q: (7) In the ring (Z, +,.), we get n {P: P non trival prime ideal in Z} %3D ....... (a) o (b) (Z, +,.)…

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Q: (c) Let I = (f(x)) be the principal ideal generated by f(x) in Z2[r]. Calculate the multi- plicative…

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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: Find the characteristic of Zm, × Zm, × · · × Zmr where m;'s are positive integers.

A: The ring ℤm1×ℤm2×...×ℤmk is a ring with unity 1,1,1,...k times Now, to find characteristic of…

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

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

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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: 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: 6. Find two ideals I and I2 of the ring Z such that a. I1 U 12 is not an ideal of Z. b. I U 1z is an…

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Q: Find all distinct principal ideals of Zn for the given value of n. n = 7

A: Find all distinct principal ideals of Zn for  n=7.

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: 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: Consider the ring R = Z. Find five different ideals satisfying the following chain: I1 ⊆ I2 ⊆ I3 ⊆…

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Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

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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: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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Q: 18. If F is a field and I is the ideal (x,xy) in F(x,y), then there are at least three dis- tinct…

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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 R=(Z9,+9,9). Find 1. Char(R) 2. Nilpotent ideals of R 3. Prime ideals of R.

A: Solution

Q: 1. An integral domain D is called a principal ideal domain if every ideal of D has the form (a) =…

A: Using Definition of Principal Ideal Domain we have to prove that Z is a Principal Ideal Domain.

Q: (3) The ring (4Z, +,.) has the following prime ideal.. (a) ((0), +, .) (b) ((8), +,.) (c) ((12),…

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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}?…

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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: 19. Find all distinct principal ideals of Z, for the given value of n. а. п 3D 7 b. n = 11 с. п %3D…

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Q: (6)lf K is a simple ring, then K is a Jacobsen radical ring. T O F

A: Please see the attachment

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

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Q: Theorem 15.2.10 Let R be a principal ideal ring and a, b e R not both zero. Then a and b have a gcd…

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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: 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.

Q: Let I be the ideal generated by 2+5i in the ring of Gaussian integers Z[i]. Find a familiar ring…

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Q: 38 Prove that I elements are in Z[i]/I? What is the characteristic of ZJi/I? (2 + 2i) is not a prime…

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Q: (7) In the ring (Z,+,.), we get n{P:P non trival prime ideal in Z} = ...... (a) o (b) (Z, +, .) (c)…

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

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