if the following ideals are Maximal, Prime but Prime. You do not have to provide reasons i. (the ideal generated by x) ii. <²> (the ideal generated by x²) iii. <2,x> (the ideal generated by 2 and x) iv. < x,x+1> (the ideal generated by x and (b) Are the following rings PIDs, a UFD but not a You do not have to provide reasons i. Q[x, y] (the ring in two indeterminates ove numbers) ii. Zp[x] (the polynomial ring in one indeterm integers modulo a prime p) iii. Z [√-5]

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: and Prove: an ideal (1,+,) of (R,+,) is the interseetion of prime ideals only if a? E I implics a E…

A:

Q: im: A proper Ideal I is prime if and

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Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

Q: а. Let a, b, c be three elements of a Euclidean ring R and (a, b) = 1. If a/bc then show that a/c.

A:

Q: your answer fully. (i) If I and J are prime ideals in R then In J is a prime ideal in R. (ii) If I…

A:

Q: 9.48. Find all prime ideals in each of the following rings. 1. Z10 2. Z50

A: Find all prime ideals in each of the following rings:1 ℤ102 ℤ50

Q: The ideal I Ð J is a prime ideal of Z Ð Z if O A. I = J O B. I = 0 and J is prime in R2 O C. I = Z…

A:

Q: Show that the set of nonunits in Z8 is an ideal of Z8

A:

Q: If (R,+,.) be aring with identity and A is : ideal of R, then the identity element of R/A is * O A+1…

A:

Q: 1. Prove that (Q, +, ·) is ring, where Q is the set of rationales numbers. 2. Prove that (Z, +, ·)…

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Q: Problem 4 Let I = (6, 15, 27) be the ideal in Z generated by 6, 15, and 27. Since every ideal in Z…

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Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A:

Q: Ex: tet R = {o, a,b,c} a G O a b ( وه هه 2 2 G 6 ( 0 a C b a O a b a b 2/2 ( в с @ Show that (B,…

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Q: (x), J = (x + 1) en K := (x²020 + x) in the polynomial ring R[x]. Rewrite the ideal I · (I + (J^ K))…

A: Aim: To define the principal ideals and in the polynomial ring . And to define as principal…

Q: 8. If 1 and 2 are two ideals of the ring R, prove that the set 1₁+= {x+yx Eh, yЄh} is an ideal of R…

A: To prove that the set I1​+I2​={x+y∣x∈I1​,y∈I2​}  is an ideal of the ring (R) and contains each of…

Q: 20. Suppose R is a commutative ring and [R]= 30. If I is an ideal of R and |I| = 10, prove that I is…

A:

Q: (b) U = {q(x) € Z3 [x] : x²+x+1 is a factor of q(x)} in Z3 [x].

A: As asked we shall solve only sub-part b

Q: If R = Z3 x Z4 and a = (2,2), which is in R. How do you find the elements of the principal ideal…

A: Given that ??R=Z3×Z4 and a=(2,2)∈R.We need to find the elements of the principal ideal generated by…

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: Let I,J,k be ideas of non - commutative division sing The or dispoove -IkJ is of R. Poove idea

A: Given: To explain the given statement as follows, We know I, J, K are the ideals of non-cumulative…

Q: What is the difference between the notation and {3} .What is the difference between the notation…

A: Here the notation &lt;3&gt; is represent ideal of Z generated by 3      i.e &lt;3&gt; = 3Z is ideal…

Q: 17. Let H and K be ideals of a ring R. Show that HNK is an ideal of R.

A: H and K be ideals of a ring R    as per the guidelines we can solve only one question if there is no…

Q: 17. Let H and K be ideals of a ring R. Show that HNK is an ideal of R.

A:

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: 1. Let R be a PID (principal ideal domain) and let a, b E R be non-zero, non- units. Recall: an…

A: Given That :  R be a principal ideal domain and let a , b ∈R be a non zero, non-units. To Prove : a)…

Q: 9. Let 11 and 12 be ideals of the ring R. Prove that the set = {a,b,+ a2b₂+ is an ideal of R. +abla,…

A: To prove that I1​I2​  is an ideal of the ring  R , we need to show that it satisfies the properties…

Q: show by of R then IUI is not ideal example if In and Iz are icdeals an

A:

Q: Every ideal in Q(i√2) [x, y, z] is finitely generated, where Q(i√2) is (isomorphic to) Q[v]/(v² +…

A: The statement is false. Every ideal in the ring ?(i√2)[x, y, z] is not necessarily finitely…

Q: For each the following, write TRUE if the statement is always true. Otherwise, write FALSE. No need…

A: Please follow the next step.

Q: 7. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A: Prime Ideal:- An ideal P of a commutative ring R is prime if it has the following two properties:…