Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers). C.

Q: Let R= Z/5Z, the integers mod 5. The ring of Gaussian integers mod 5 is defined by R[i] = {a+ bi :…

A: Def zero DevisorIn a ring R , an element a∈R will be zero devisor if there exists a non-zero b∈R…

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…

Q: Let R= Z/5Z, the integers mod 5. The ring of Gaussian integers mod 5 is defined by R[i] = {a+ bi :…

A: A non-zero commutative ring with unity having atleast two elements and without zero divisor is…

Q: There are.... Polynomials of degree atmost n in the polynomial ring Z, (x O none O5+5^n O 5^(n+1) O…

A: The general form of the polynomial of degree n is  Pn(x)= a0+a1x+a2x2+...+anxn .

Q: disprove that the is smallest non- Prove commutative ring oY of order 4-

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Q: if A and B are ideals in a ring R such that A intersect B ={0}, prove that for every a in A and b in…

A: Let A and B are ideals of a ring R such that A∩B=0

Q: Let n be a positive integer. Show that there is a ring isomorphismfrom Z2 to a subring of Z2n if and…

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Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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Q: Prove that if a is a ring idempotent, then an = a for all positive integers n.

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Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.

A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…

Q: 4)If D is a ring of integers module 13. Then rad (D) = {0}. От OF O O

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Q: Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 forall n >= 1.

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Q: In the ring Z[V5], the element 3+ V5 is O Irreducible O reducible O prime O unit

A: Option A

Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: Let R be a commutative ring. If I and P are ideals of R with P prime such that I ¢ P, prove that the…

A: The ideal quotient of P and I is P:I=x∈R : xI⊂P which is again an ideal of R. Given that P is a…

Q: - Prove that, if I is an ideai of the ring Z of integer numbers then I=, for some nɛZ'U{0}

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Q: Let a and b be elements of a ring. Prove that (-a)b = -(ab).

A: Solve the following

Q: In the ring Z Z,1= {(a,0)|a € Z} is: O prime not maximal O maximal ideal O neither prime nor maximal

A: The given ring is, Z⊕Z and I=a,0a∈Z

Q: Let n be an integer greater than 1. In a ring in which xn = x for all x,show that ab = 0 implies ba…

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Q: (5) I is maximal ideal of ring R if and only if R = (a, I) for any..... a in R O a in I a in R-{I} o…

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Q: in the ring a such that (a)=12) +<18) of the integers, Z, find a positile integer

A: In the given ring of integers Z, we have to find a positive integer a such that  &lt; a &gt; = &lt;…

Q: In the ring Z O Z, I = {(a,0)|a € Z} is: O prime not maximal maximal ideal neither prime nor maximal

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Q: In the ring Z4 O Z4.1= {(0,b)|b € Z4} is: O maximal ideal prime not maximal O neither prime nor…

A: Given ring is

Q: In the ring Z Z , I = {(0, b)|b € Z} is : maximal ideal prime not maximal neither prime nor maximal

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Q: Show that in a Boolean ring R, every prime ideal P is not equal to R is maximal.

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Q: Show that the ideal of (5) in the ring of integers Z is the maximal ideal.

A: An ideal A in a ring R is called maximal if A ≠ R and the only ideal strictly containing A is R. In…

Q: The set of all zero divisors of the ring Z6 is اختر احدى الاجابات O {2, 3, 4} O (1, 3, 5) O {1, 2,…

A: Z6={0,1,2,3,4,5}Since 2 · 3 ≡6≡ 0 (mod 6) and 3 · 4 ≡12≡ 0(mod 6)However, 1 and 5 are not zero…

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: с. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

A: (C). Prove that neither 2 nor 17 are prime elements in Zi.   Note : An integer a+ib in Zi is a prime…

Q: Show that for every prime p there exists a field of order p2.

A:

Q: Let R be a ring and a=a for all a'e R, Then commutative. prove that R is

A: First we notice that x3=x for all x∈ℝ, so that means 2x3=2x and thus 8x=8x3=2x and so 6x=0. Thus…

Q: In the ring ZO Z, 1 = {(a,0)|a € Z} is: O None of these O prime not maximal O neither prime nor…

A: We know that quotient is integral domain iff ideal is prime.

Q: In the ring Q[x], every prime ideal is maximal. O True False

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

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Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

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Q: There are. Polynomials of degree atmost n in the polynomial ring Z, (x). *** O 5+5n O none O5n O…

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Q: In the ring ZO Z , 1 = {(a,0)|a E Z} is: neither prime nor maximal O maximal ideal O prime not…

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Q: Find the integral basis and discriminant of the number field K = Q(/=p), where p is a prime such…

A: It is given that

Q: In the ring of integers modulo n, (Z„ +, ·) prove that m e Z, is a zero divisor e (m, n) > 1.

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Q: C. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

A: NOTE:Hi! Thank you for your question. Since,we only answer 1 question in case of multiple question,…

Q: There are . Polynomials of degree atmost n in the polynomial ring Z,[x]. O 7+7^n O 7^(n+1) none O…

A: Option B

Q: The set of all zero divisors of the ring Z6 is اختر احدى الدجابات O (2,3,4) O (1,3,5) O (1,2,3,4,5)…

A: The set of all zero divisors of the ring Z6 is (0,2,4)

Q: An element x in a ring is called an idempotent if x2 = x. Prove that  the characterstic of R is 0 or…

A: An element x in a ring is called an idempotent if x^2 = x

Q: In the ring Z Z , I = {(0, b)|b € Z} is maximal ideal prime not maximal neither prime nor maximal

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Q: (b) if R is commutative and has no ideals other than {0} and R, then R is a field.

A: 4.b) Given that R is commutative with with unity has no ideal other than {0} and R.

Q: 2) Let P + Q be maximal ideals in a ring R and a,b elements of R. Show that there exists c E R, such…

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Q: Label each of the following statements as either true or false. The characteristic of a ring R is…

A: To determine whether true or falseThe characteristic of a ring R is the positive integer n such that…

Q: There are .. Polynomials of degree atmost n in the polynomial ring Z,[x O 7^n O 7 + 7^n O 7^(n+1) O…

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Q: In the ring Z4 O Z4,1 = {(0,b)|b € Z4} is: O neither prime nor maximal O prime not maximal maximal…

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Q: Let A be a ring and suppose that for some even positive integer n we have a" =a. Then, for every a E…

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