1) The proper ideals of 7z is a) { o, 2, 43 b) fo,1, 3,5} c) to}i d) [o,1.2, 3, 4.5,6 }

Q: Let R = {6 la, be Q}and let p:R – Q be defined by : 0( D = 1) is a ring a) Homomorphism. b)…

A: Here, first we will check the the isomorphism property and then we will find the kernel , See the…

Q: a) Find C and k that show 8x? + 5× is O(x?). Show your algebra reasons. b) Use proof by…

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Q: In the ring Z:[x] 1- The inverse of x² is a) x+1 b) x2 +1 с) х d) x2 2- The value that replaces x is…

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

Q: 5. The number of elements of Z3[x] / is а) 3 b) 6 c) 8 d) 9 O a) O b) O c) d)

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Q: = a . Let S = { la, b e R} and let p:S → R be defined by : Ø( D 1) is a ring

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Q: The ring Z,2, has exactly-- --maximal ideals O 2 3.

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Q: Let S = {: la, be R}and let o:S R be defined by: = a . 1) o is a ring a) Homomorphism. b)…

A: Since you have asked multiple subparts question, we will solve the first three subparts of the…

Q: a) Find the Q-conjugates of 5. b) Find the Q-conjugates of √2+√5. c) Find the Q-conjugates of…

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

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Q: Suppose x,y,z are mutually coprime positive integers that solve x2+y2=z2. a) Show why exactly one…

A: Basic properties of modular arithmetic and gcd will be used in solving this problem. The detailed…

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

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Q: a) If U = {2,4,6 ... 10} is a universal set and A = odd number B = Prime number, C = Prime number…

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Q: The number of zero divisors of the ring Z4 O Z2 is a) 11 b) 5 c) 7

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Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: 5) A vng R is Called Boolean ving if.... a) x: y? Vx, Ye R b) x xy Vx, yeR c) None d) x:y vx, yeR

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

Q: Let 2Z and 5Z be ideals of Z. Compute 2Zn5z. Prove that 1 E 2Z+5Z.

A: We use basic concepts of ideals to solve this problem.

Q: 1. The product of 3x²_5x² + 3 and 2x + 5x – 4 in z,[x]/ is а) 2x b) 2x + 1 c) 2x + 2 d) 2x + 3

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Q: Let R = { la, b e Z}and let o:R- Z be defined by : 1) ois a ring a) Homomorphism. b) Isomorphism c)…

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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: 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: = {I6 Tla, b e Q}and let p:R - Q be defined by : 0( 1) ois a ring a) Homomorphism. b) Isomorphism c)…

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

Q: Prove that

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Q: 7) Prove te follo aing theorem 18 A and B ve madrices cof siees such that ore odition is defined…

A: Suppose, A and B are matrices of same size such that,    Where,   Now,…

Q: d) None 4) Ker (6) is: a) Prime ideal not maximal b) Maximal ideal not prime c) Both maximal and…

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Q: 1) The of 77 is proper ideals a) { o. 2, 43 b) fo, 1, 3,5} c) { o}i d) {0,l,2, 3, 4,5,6 }

A: Given :   Proper ideal of z7

Q: Let R = %3 { la, b e z}and let p:R - Zbe defined by : 0(1 ) = - . 1) is a ring a) Homomorphism. b)…

A: The solution is given by using definitions of homomorphism, isomorphism and kernel as follows

Q: = 36 ala,beZ}and let ø : R → Z be a mapping defined by : [a b -6. b

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

Q: 17.6 Find a maximal ideal in a) Zo;

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Q: P Suppose z = x + iy and w = u + iv. a) Prove that z+w = z+ w (i.e., the conjugate of the sum is…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: 2. Prove that the Cartesian product N Nis countable

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Q: Give 3 different examples of norms on R

A: In (1) and (2) same question is written. Here solving 3 different examples of norm on R. If you want…

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

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Q: 1. In Z[i], 1+2i and ___ are associates a) 2+i b) -2-i c) 2-i d) 1-2i e) none of the above 2. What…

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Q: 3) Given two prime numbers p and q, p p is invertible module pq What can we say? A) Both…

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Q: 10. When dividing x* + 3x? + 2x + 1 by x² + 2x + 3 in Z5[x], the remainder is а) 0 b) 2 c) 3 d) 4 a)…

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Q: 1+xr 1+x* a) b)

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