Exercise 36. Let A,B, and C be ideals of a commutative ring R. Prove the following: (1) A.(B.C) (A.B).C, (2) A.B B.A, (3) A.BCANB,

Q: IF J is nil left ideal in an Artinian ring R, then J is

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Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12}.

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Q: D Let I be an ideal of ring R Such Hhut when cver R is Commutakive with idlentily then so is the…

A: Let R be a ring and let I be an ideal of R. We say that I is prime if whenever ab ∈ I then either a…

Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +, .) (c) ((0),…

A: The given question is solved with explanation below.

Q: Q2) Let (M₂ (R), +..) be a ring. Prove H = {(a) la, b, c €. = {(ab)la, b, c € R}is a subring of (M₂…

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Verify in the following statement involving the ideal generated by (a1, a2 , ... , ak ) in the ring…

A: To show that 2,3=ℤ Let 2,3=2x+3y: x,y∈ℤ -------(1) gcd2,3=1 , Thus 2m+3n=1 for some…

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

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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: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak )) in the ring…

A: To prove that: (2, 5)=ℤ

Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

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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 ring, a, b e R with a + 0. Then the equation ax = b has a solution in R.

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

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

Q: 9. Suppose that (R,+, .) be a commutative ring with identity and x E rad R, then ..... .... (a) (x)…

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Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

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Q: The ring Z37 has ... idempotent elements O 10 О 36 O 2 О 20

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Q: 9. Suppose that (R,+, .) be a commutative ring with identity and xE rad R, then (a) (x) = R (b) 1-x…

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Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

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Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12).

A: Given problem Given that   In a ring Z , We have to find the Least common multiple of { 2,3,8,12…

Q: The characteristic of the ring Z3XZ6 is 9. Select one: O 3 O None O 18 O 6

A: The characteristics of the ring is defined as  the least positive integer n such that na=0 for all…

Q: Rp { -b a | a, b = Zp}

A: It is given that,  Rp = ab-ba : a, b ∈ ℤp We have to prove that,  Rp is a commutative ring with unit…

Q: 4. Prove that a zero divisor in a ring cannot be a unit.

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Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

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: Show that the subring of an artinain ring may not be artinian???

A: According to the given information, it is required to show that subring of an Artinian ring may not…

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: Let R, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…

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Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0),…

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

Q: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…

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Q: Q Show that Z12/17) is a quatient ring.

A: We have to show that ,   ℤ12(4¯)  is a quotient ring.

Q: Prove that the number i5 is not reversible in the ring Z[V-5]

A: Here we show that isqrt(5) is not reversible in the ring Z[sqrt(-5)].

Q: If Ris a ring with identity and a is a unit, prove that the equation ax = has a unique solution in…

A: Let R be a ring with identity and a∈R be a unit. Prove that the equation ax=b has a unique solution…

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…

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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: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

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

A: Ideals of the ring Z12

Q: 1- Let (R,+,-) be aring which has property thut a=a, Ua ER.prove thatR is Commutabive ring (Every…

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Q: Which one of the following is not a commutative ring? (a) (Q. +, :) (b) (Z5, O5, 85) (c) (Z12, O12,…

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Q: Suppose I,J be ideals of a commutative ring R. Prove that IJ cInJ.

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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: 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 Zpq?r has exactly------------maximal ideals O 3 2

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

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: Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.

A: Proof. Let R be commutative. Then ab = ba for all a,b ∈ R.

Q: s{a+bV2:a, b e Z } under addition and multiplication a ring? Justify. Is it a mmutative ring?

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Q: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak ) in the ring…

A: we have to prove,(10,15)=(5)

Q: Theorem 2.7. IfI, I, I2, . .., I, are ideals in a commutative ring R, then: (i) Rad (Rad I) Rad I;…

A: The solution for the above question is as shown below.