Let R denote the ring of integers modulo 48 and S denote the ideal (0, 16, 32). The order of the quotient ring R/S is b. 16 d. 6 a. 45 c. 32 e. 49

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

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Q: Let n > 2 be an integer. Show that Z/Zn is a field if and only if n is a prime number.

A: Let A=1¯,2¯,3¯,...,n-1¯ we show for ⇒ Let ℤℤn is a field⇒∀a∈A,∃x∈A such that ax=1¯ax≡1modn has a…

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Q: Let Q[√c] denote the set of numbers a + b ⋅√c, where a,b ∈ Q for some c ∈ Z. I must prove that Q[√c]…

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Q: Q:- Consider a ving R={o,a,b,c,d, e,f.g} defired by folowing the cayley's tables. elf ef +) a e a d…

A: Solution: The given ring is R=0,a,b,c,d,e,f,g We have to calculate the characteristic of the ring R.

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Q: Describe the elements of the extension Q(4√ 2) over the field Q(√2).

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Q: xx+24=5x² fpyonbouy of anyos

A: We have xx+24≤5x2 Consider:…

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A: 3

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Q: O) be the smallest ideal of R that contains a. I6 Ris a commulative Ring uith unity , show That…

A: Given, a be the smallest ideal of R that contain a.If R is a commutative Ring with unity,   To prove…

Q: Let R = Z/3Z, the integers mod 3. The ring of Gaussian integers mod 3 is defined by R[i] = { a + bi…

A: Solution

Q: L. Let h (x) = f (x) g (x). Find a. h' (1), b. h' (3), and с. h' (4).

A: According to the given information,

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Q: Let be the following set of rational numbers: Consider the algebraic structure (R2, +, ·), where +…

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Q: - show that Zo find the Zero following rings 071 2 78 ax Rings. divisor of Find the Zero divisor of…

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: 1. Find all units in Z[V-3]. Justify your answer. 2. Prove that 2 is an irreducible element in…

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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

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Q: The set None.d o Both a and b.c O -R is an ideal of any ring R.b O 1-{0} is an ideal of any ring R.a…

A: solve the following

Q: 1. If S is a subring of R, then S[a] is a subring of R{x].

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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: 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: Which of the following elements of the following rings are units? Select all that apply: O The…

A: Result 1: Let (F,+,∘) be a field whose zero is 0F and whose unity is 1F Let F[X] be the ring of…

Q: 3. Let I and J be ideals of a ring R and let b, c and f be elements of R. Prove that if b = f(mod I)…

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Q: Q23. Find the maximal ideals of Zg.

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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…

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Q: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…