) Let Z3[i] = {a+ bi | a, b E Z3}. Show that Za[i] is isomorphic to Zs[a]/ (a² + 1). ) Is the ideal (r² + 1) maximal in Z,[r]? Why or why not?

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

Q: 7. Find all distinct principal ideals of Z14

A: Since Z14 are cyclic group with respect to addition so all ideals of Z14 are principal ideals

Q: Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

A:

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…

A:

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…

A:

Q: Let BCR be a subset of R. Prove that there is an ideal I of R such that B n I has at most one…

A: To show that ideals I and J  exist satisfying conditions 1) and 2)

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

A:

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

A:

Q: Show that A = {xf(x)+ 2g (x) : f (x), g (x) e Z [x]} is not a principal ideal of Z [x] and so Z [x]…

A:

Q: Prove that I = is not a prime ideal of Z[i].

A: Solution :-

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

A:

Q: 8. a. Prove that I[x] = {do + ax + + a,x"|a, = 2k, for k, E Z), %3D %3! ... the set of all…

A:

Q: Find all maximal and prime ideals of Z10-

A:

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

A:

Q: The ring Zpq²r has exactly-----------maximal ideals 1 2 3

A: 3

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

A:

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

A:

Q: Is cartesian product of prime ideals of a ng also a prime ideal? prove or give a counter example.

A:

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

A:

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

A:

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

A:

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

A: Yes it is true .

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

A:

Q: - Let R=(Z,t9 ,.9). Find 1. Char(R) 2. Nilpotent ideals of R 3. Prime ideals of R.

A:

Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

A:

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.

A:

Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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

A: Solution

Q: The ring Z,2, has exactly--- --maximal ideals 1 2.

A:

Q: Find all the principal ideals of the ring (Z6, O, 0 0 ).

A:

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

A:

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: 7. Show that the splitting field Q(√2, √3) of (X2-2)(X2-3) is a Galois extension of the rational…

A: If there exist a polynomial of degree 4 generated by this splitting field, then we say this is an…

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]

A:

Q: Prove that the set of all polynomials whose coefficients are all evenis a prime ideal in Z[x].

A: Assume A to be the subset of X[x] with all even coefficients.For∑i=0maixi∑j=0nbjxj=∑i=0maxm,nai−bixi…

Q: The ring Z-[i] has no proper ideals

A:

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.