Char(2Z/6z) is : 4 Оз None of these O 2 The ring Zpg?r has exactly-------------maximal ideals O 4 2 Оз 1

Q: I = {a + bi | a,b∈ℤ, a≡b (mod 2)} Show that the set is a prime ideal

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Q: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is 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: 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: 3. In the ring (Z12, +12,-12 ), the following ideal is maximal (a) (2) (b) (3) (c) (a) and (b) (d)…

A: An ideal of any ring is maximal ideal if it is the largest ideal. Suppose P and Q are two ideals of…

Q: Find the number of ideals in Cx,y that contains both the polynomials y – (x – 1)³ +1 and p(x) in…

A: We have to find the number of ideals in ℂx,y that contains both the polynomials y-x-13+1 and px in…

Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: Prove that {0R} is not a prime ideal in Z12.

A:

Q: Find all maximal and prime ideals of Z10-

A:

Q: Show that N= (3x, y)|x, y ∈ Z  is a maximal ideal ofZ×Z. Note that you must show that it is indeed…

A:

Q: Consider the ideal N= of the ring Z24. How many elements does the quotient ring Z24/N have?

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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, +,.)…

A:

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

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Q: 22 (b): let A and B two ideals with unity of Commutative A+B= IR ving IR Such that Prove A.B = ANB.…

A:

Q: 3. Are the rings 2Z, 3Z isomorphic. Any isomorphism is linear for addi- tion, so it has the form f…

A: Ring Isomorphism: Letus consider two rings R and S.Let us consider the mapping  f: R→S. Then it is…

Q: 2) A ving having no proper ideals Called single Ying is D True O Fulse i

A: A ring having no proper ideal is called single ring. to determine whether the statement is true or…

Q: Q4: (A) Prove or disprove each of the following 1. Every prime ideal is maximal. 3. The cancellation…

A:

Q: true or false In the domain of Gaussian integers Z[i], the element 17 is Irreducible

A:

Q: Suppose that r and y are integers that satisfy y = x² + 2 a) By considering congruences modulo 4…

A: Answer for sub question a: Note: Cube of any integer is congruent to either 0 or 1 or 3 modulo 4.…

Q: 4. Give addition and multiplication tables for 2Z/8Z. Are 2Z/8Z and Z4 isomorphic rings? Concents

A: Let, nℤ be an ideal of a ring ℤ. Then the additive and multiplicative Cosetof nℤ can be defined as,…

Q: Is the polynomial x2 – 4 in the ideal generated by the polynomials x + 22 – 4x – 4, x – 22 – 4x + 4,…

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Q: (4) Let Z12 be a ring of integer modulo 12. Then there are.....maximal ideals of Z12- O (i) 4 O (ii)…

A:

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

A:

Q: Explain why (or prove) {∑nk=3 akxk|ak ∈ R} is not a prime ideal in R[x].

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Q: 9- Z is Principal Ideal Domain. a) True b) False O a) True O b) False

A:

Q: (10) Let I = (4) be a principal ideal of integer %3| ring Z, Then I is. . ideal * Primary Prime O…

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Q: Let an ideal I = {(a, 0): a ∈ Z} of the ring R = Z ⊕ Z be given. I, R is a prime Is it the ideal

A: To prove: I is an ideal and it is prime ideal of R. Here, R=Z⊕Z Let ,I=a,0:a∈ℤ A non empty subset I…

Q: Select one: O 5Q is a maximal ideal in Q O 7Z is not maximal ideal in Z O None O {0} is a maximal…

A:

Q: n- tyivial 20,3 } {0,2,43 are 1) The ideals of and non-

A: The non-trivial ideal of Z6 {0,3} and {0,2,4} .

Q: I8. Let (R, 1.) be a commutative rinK with identity and let N denote the set of nilpotent elements…

A: a) Let (R, I, ·) be a commutative ring with identity. Let N be the set of nilpotent elements of R.

Q: Explain why 3Z is not an ideal in Q

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Q: find the maximal ideals

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Q: Q2) Select the correct answer for the following: (10 Marks) (1) Let I be a proper ideal of the…

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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)…

A:

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

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Q: Not prime ideal of R

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Q: → Z₂ such that y(x) = {0 if x is even 1 if x is odd momorphism because ria for irreducibility Test…

A:

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

A: Solution

Q: (B) Explain the relationship between an a) Ideal and subring. b) Prime ideal and maximal ideal. c)…

A:

Q: Q: (A) Let R =(Z12, +12-12). 1. Find all maximal ideals of R. 2. Evaluate char. (R). 3. Is R a…

A: Introduction: Rings are algebraic structures in mathematics that generalise fields: multiplication…

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

A:

Q: 8 Every quotient of a principal ideal domain is a principal ideal domain. 9. Every integral domain…

A: Given: 8.Every quotient of a principal ideal domain is a principal ideal domain

Q: 8.38. Find the characteristic of each of the following rings. 1. Z4 O Z10 2. M2(Z3)

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Q: These ideals (0), (2),(3),(5) are prime ideals of Z also all of these are maximal ideals of Z. ㅎㅎ Ⓡ

A: We can solve this using definition of maximum ideal and prime ideal

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

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Q: Let k, m, n be positive integers with k <m <n. Give a combinatorial proof of the identity k m m+ Σ k…

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

A:

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}

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.