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

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

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A: Solve the following

Q: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab ba. Then (a) (a+ b)…

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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Q: a is a unit in a ring R with unity, then a is not a zero divisor

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Q: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab = ba. Then (a) (a +…

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Q: If A and B are ideals of a commutative ring R, define the sum of A and B as: A+B = {a+b| a € A, b €…

A: Let A and B be two ideals of a ring R. If  A and B are ideals of a commutative ring of R with unity…

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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: Let R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

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A: We have to solve given problem:

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Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

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

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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…

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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: if A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (A B) + (AnC).

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).

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Q: Let a and b be elements in a ring R. If ab is a zero divisor, prove that either a or b is a zero…

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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: For any n E N, each zero divisor in the ring Z, is nilpotent.

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

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

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Q: Let R be a ring and M be an R-module. Ther

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

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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: Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides…

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Q: The number of zero divisors of the ring Z, O Zg is

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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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