3. In the ring (Z12, +12,-12 ), the following ideal is maximal (a) (2) (b) (3) (c) (a) and (b) (d) (a) or (b) (a) O (b) O (c) O (d) O 4. The ring (Z, +,.) the ideal (12) containing in the following maximal ideal .. ..... (a) (4) (b) (5) (c) (6) (d) No Choice

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

A:

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

A:

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

A: Given ring is ℤ12.

Q: give an example of An ideal in Z that is not a prime ideal.

A: The objective is to write an example of an ideal in ℤ that is not a prime ideal.

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

A:

Q: Find

A: We know that,  All ideals of Zn are kZn, Where k is divisor of n.

Q: For technical reasons, the equivalence class of a will be denoted byā in the first column, and it…

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

A:

Q: What are the maximal ideals in Z12? , O , O , O

A: Solution: Definition: A maximal ideal of a ring R is an ideal M≠R such that there is no proper ideal…

Q: One of the following is an ideal of z1: (0,3} (0,2,4,6} None (9,3,6,9}

A: From given statement

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

A:

Q: 18.1.4. Find all the units, irreducibles, and primes in the ring (Z/10Z, +, ·).

A: The ring is a commutative ring with identity but is not an integral domain. The units of the ring…

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: Find all solutions of x2 + x = 0 in the ring Z12

A: Given x2+x=0

Q: a) The idempotents Of (Z6,0,0,) are ONLY 0, b) The number 161 is an irreducible element in Z[i] c) A…

A: As per the company rule, we are supposed to solve the first three sub-parts of a multi-parts…

Q: (B) Find all maximal ideals of (Z8, +8,8).

A:

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

A:

Q: 1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т…

A:

Q: In the ring Z Z , I = {(0, b)|b € Z} is : maximal ideal prime not maximal neither prime nor maximal

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: Show that the ideal of (5) in the ring of integers Z is the maximal ideal.

A: An ideal A in a ring R is called maximal if A ≠ R and the only ideal strictly containing A is R. In…

Q: 17. a) Show that the annihilator of a semisimple ring (R, +,) is zero; in other words, ann R = {0}.…

A: A semisimple ring.

Q: Let R=Z[X], and let I be the ideal generated by x3 +2x. Find integers a, b, and c such that in R/I…

A: Given that,R = ℤx and I = the ideal generated by x3 + 2x.Now, I = <x3 + 2x> and I be the…

Q: show that Q[x]/(3x⁴+2x³+1) is a field. Here (3x⁴+2x³+1) is the principal ideal generated by a…

A: Given factor ring is ℚx3x4+2x3+1 where 3x4+2x3+1 is the principal ideal generated by the polynomial…

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

A:

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0),…

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

Q: 7) There is no proper non-trivial maximal ideals in (Z2₁, because is a maximal ideal in Z2₁, ,0 )…

A: Solution

Q: The ring Zpg²r has exactly-- ---maximal ideals 2 1 4

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

Q: 2- An example of two ideals A and B of the ring (Z12. +2-u) such thata B are: (a) A and B (c) A- and…

A: First we find all subgroup of Z12

Q: It is known that 28= {0, 1, 2, 3, 4, 5, 6.73 is a Ring. H = { 0,43 is a subring of Z8. show that…

A:

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

A:

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

A:

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

A: Solution

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

A:

Q: What are all of the distinct generators of Z17 with respect to multiplication? What are all the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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: (7) The set H={0,1,2}is a subring of (Z4, +4,.4) integer ring module 4. T OF

A:

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

A: Prime ideal sometimes behaves like a prime numbers . Let's firstly define prime ideal.

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: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

A:

Q: In the ring Z Z , I = {(0, b)|b € Z} is maximal ideal prime not maximal neither prime nor maximal

A:

Q: et [g] be an element of order 180 in Z181. Let [a] be a non-zero element of Z181, what are the…

A:

Q: What is the principal ideal (3) of z8 = {0,1,2,3,4,5,6,7}? Is it not just z8?

A:

Q: 17. Given that (I, 1.) in an ideal of the ring (R,+,), show that a) whenever (R,1,) is commutative…

A: Definition Two-sided Ideal of a ring : An ideal I of a ring R is called a two-sided ideal if it…

Q: (3) (a) Suppose a ring R is a finitely generated algebra over a field k. Prove that the Jacobson…

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: Find all maximal ideals in: (a) Z10 (b) Z21

A: (a) Maximal ideal of Z10= (i) {0,2,4,6,8}   (ii) {0,5}

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

A: