The ring Zpq?r has exactly------------maximal ideals O 3 2

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: Q2. Recall the ring of infinitesimals C[e] that was introduced in the first lecture. Find all units…

A: Cε=Rε∈Cε |  R ε is polynomial in ε Let R be any Ring. 0≠x∈R is said to be unit if there exist…

Q: In R[x], prove ideals (x3 – 1, x7 – 1) = (x – 1, x – 1).

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Q: Show that A is ideal of A+B? IS an

A: According to questionWe have to show that A is an ideal of A+B

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: 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: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: The number of zero divisors of the ring Z, Z5 is О 1 O 5

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

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: The ring Zpq²r has exactly-----------maximal ideals 1 2 3

A: 3

Q: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

A:

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

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

Q: If A and B are ideals of a ring, show that the product of A and B, AB = {a,b, + a,b, +· · · + a,b,l…

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Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

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

A: Solution:

Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

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: 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: EN is an ideal of R.

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Q: The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3

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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: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

Q: The ring 5Z is isomorphic to the ring 6Z True O False

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

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Q: The rings Z and 5Z are isomorphic.

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Q: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

A:

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

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

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

A:

Q: 5. Let A and B be two ideals of a commutative ring R ith unity such that A + B=R. Show that AB =…

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Q: Give an example of ring elements a and b with the proerties that ab = 0 but ba does not equal 0.

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

A: In the question it is asked to find whether <x2-1> is a principal ideal.

Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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Q: 9. The number of ideals in the ring (Zis.+1s1s) is: (a) Two (b) Three (e) Four (d) Five.

A: The positive divisor of 15 are: 1,3,5,15

Q: Give the following theorem (without proof): If (R, +, ·) is a ring, and S C R then what is the…

A: That's easy. Thumb up. Have a great day!!!

Q: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

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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: If I1 and I2  are two ideals of the ring R, prove that Ii n 11 ∩ I 2  is an ideal of R.

A: Given  I1 and I2  are two ideals of the ring R To prove : I1∩I2 is an ideal of R.

Q: Show that for any positive integer n, gl,(C) can be written as a direct sum of two ideals, one of…

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Q: let I be a non-Reso ideal.

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Q: is union of two ideal rings R, an idea of R? prove or give counter example

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Q: 5. Let A and B be two ideals of a commutative ring R with unity such that A +B = R. Show, that AB=A…

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Q: The ring Zp2, has exactly-----------maximal ideals

A: