3. In the ring (Z,2, +12,-12 ), the following ideal is maximal ........ (а) (2) (b) (3) (c) (a) and (b) (d) (a) or (b)

Q: 4. The ring (Z, +,.) the ideal (12) containing in the following maximal ideal ... (a) (4) (b) (5)…

A:

Q: If Z is integers ring, then each non zero element in a proper ideal of Z is......... O (i)…

A:

Q: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

A:

Q: Consider the ring ℤ12. List the elements of the following principal ideals: (i) ⟨4⟩  (ii) ⟨9⟩…

A: Given ring is ℤ12.

Q: Let S = {0, 2, 4, 6, 8} CZ10. Is S an ideal of Z10?

A:

Q: 5. Let R be a ring and r1,., r, € R. Prove that the subset (r1,., "n) = {Airı + Anrn | A1,. An E R}…

A:

Q: (4)Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to…

A: Given that, K=Z12 and ideals I and J are such that I=([4])=4Z12={0,4,8} J=([6])=6Z12={0,6} Now,…

Q: 2. Let Z [√2] = { a+b√2 la,be Z}. Define addition and multiplication on Z[√2] as follows: (a+b√2)+(c…

A:

Q: 17.6 Find a maximal ideal in b) Z12;

A:

Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

A: The objective of the question is to determine whether the union of two ideals I and J of a ring R is…

Q: et J be the ideal of R = Cla, b, c, d] generated by a²b, b³c, cd, d5 a. a) Write A = V(J) as a union…

A: Given: The Given Ideal of R=Ca,b,c,d generated by a2b,b2c,c4d,d5a. To write A=VJ as a union of…

Q: The number of reducible monic polynomials of degree 2 over Zz is

A: The number of reducible monic polynomials of degree 2 over Z3 is

Q: (5) I is maximal ideal of ring R if and only if R = (a, I) for any..... a in R O a in I a in R-{I} o…

A:

Q: Which of the following are ring (i): f: (2Z, +,-) → (Z, +,-), (ii): ƒ: (Z8, +,-) → (Z4, +,-), - 2x -…

A:

Q: a.) Ker(p) and p(24) for p : Z → Z7 such that Ф(1) %3D 4 b.) Ker(p) and p(16) for p : Z → Z10 such…

A:

Q: and Prove: an deal (7, F,') of (R, -+,) is the intersection of prime ideals only if a2 E I implies a…

A: Let R be ring and Let I be an ideal of R.

Q: In the factor ring Z[i]/ the element 12 – 5i+ is equivalent to

A: Given that Z[i]/<3+i> The objective is to find the equivalent for 12-5i+<3+i>.

Q: Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])

A: Let * be integer ring module 12.  Let  I=4J=6

Q: (1) Let I be ideal and a does not exist in I, then I is a subset of (a, I). T O F

A:

Q: 5. In the ring (Mat,x2(Z), +,.), if A € Matzx2(Z) then the ideal (A) = (a) {B. A: BE Matzx2(Z)} (b)…

A:

Q: 6. Indicate whether each of the following statements is True (T), or False (F). Explain your…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 19. Prove: an ideal (I, +, ·) of (R,+, ·) is the intersection of prime ideals if and only if a? E I…

A: Let I,+,· is an ideal of R,+,·.  To prove that I is the intersection of prime ideals if and only if…

Q: In Z[√ 5], prove that both 2 and 1 + √ 5 are irreducible but notprime.

A: Consider the norm function N(a+b√5)=a²−5b², which is multiplicative in Z[√5] , in the sense that…

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

A:

Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D

A: The term radical is used when we think about ideals and when we talk about ideals definitely…

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: