Consider the ring of polynomia! Q [ Z] 2 -1 €Q[x] is aprincipl ideal 5

Q: Show that in the factor ring Z[x] /(2x+1), the element x+(2x+1) is a unit.

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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: 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 R be a ring with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.

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Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: Find all pro pes ideals of the ring (Z24s +24 '24)·

A: All Ideals of z_24 are 8..... And proper ideals are 6

Q: Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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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: There are. Polynomials of degree atmost n in the polynomial ring Z, [x]. none 5^(n+1) 5^n 5 + 5^n

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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: There are.... Polynomials of degree atmost n in the polynomial ring Z, [x]. 5^n O 5+ 5^n Onone…

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Q: The ring Z3[i] has no proper ideals aya Math ele haw

A: O have proved the general result for arbitrary field.

Q: There are . Polynomials of degree atmost n in the polynomial ring Zs (x]. O 5 + 5^n O 5^n O 5^(n+1)…

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Q: 16. Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z*; n2 1} Show…

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Q: a) The idempotents Of (Z6,0,0,) are ONLY 0, b) The number 161 is an irreducible element in Z[i] c) A…

A: As per the company rule, we are supposed to solve the first three sub-parts of a multi-parts…

Q: Show that if R is a ring with unity and N is an ideal of R such that N R, then R/N is a ring with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

Q: The ring Z is isomorphic to the ring 3Z O True False

A: Solution:

Q: The ring 5Z is isomorphic to the ring 6Z

A: Since the z is same for both and the multiplier is different i.e. 5 &6.

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: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Let R be a commutative ring with unity, and let c ER be a fixed element. (a) Prove that the set A =…

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Q: 2. Let R be a ring: The center of R is the set 3XER: ax= xa vae R? Prove that the center of a ring…

A: Let R be a ring .We have to show that centre of ring is a subring of R

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

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Q: b) Prove that, if S is a ring with characteristic 0, then S infinite.

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

A: General form of the polynomial of degree n is  a_0+a_1x+...a_nx^n.

Q: Is the mapping from Z5 to Z30 given by x --> 6x a ring homomorphism?Note that the image of the…

A: Given a mapping,

Q: (17) Prove that the ring Zm Xx Z, is not isomorphic to Zmn if m and n are not relatively prime.

A: We have to prove given property:

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: 2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime…

A: Given R be a commutative ring with unity and if I is a prime ideal of R. Then we have to prove that…

Q: The rings Z and 5Z are isomorphic.

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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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…

Q: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

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Q: There are ... Polynomials of degree atmost n in the polynomial ring Z3[x]. none 3^n O 3^(n+1) O 3 +…

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Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: a. Let R and S be commutative rings with unities and f:R -S be an epimorphism of rings. Prove that S…

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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: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…

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Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…

Q: The ring 3z is isomorphic to the ring 5z True False

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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…

A: