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: 3. Let R. be a commutative ring with ideals L and M. Give an example of when LMLn M.

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Q: a Let Z[√2] = {a+b√√2 | a, b € Z} and let H = { b Show that Z[√2] and H are isomorphic as rings. 2b…

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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: Que 405: Let S be a prime ideal of R and suppose that a product 1a2a3... an, of elements of R,…

A: Let S be a prime ideal of R and suppose that a product , of elements of R, belongs to S. Then prove…

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

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Use the Maclaurin series e- -Σ. -iz -e and the definition sin(z)= %3D n=0 n! to find the Maclaurin…

A: In this question, we use the exponential series to expand f(z)= sin(z) as the Maclaurin series.

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…

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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: (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: (b) Let I, J be ideals (two-sided) of a ring R and check as to whether In J is an ideal (two-sided)…

A:

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…

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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: 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: 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: 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: Which one of the following is true? In the ring Z6, if x = x then just x = 0 or x = 1 In the ring…

A: Option (3) is correct. In the ring Zp, if xy=0, then x=0 or y=0.

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:…

Q: a ring and až a for prove that R is. Let R be all aER Then commutative (6) Let A and B. commutati ve…

A: