An ideal A of a commutative ring R with unity is said to be finitelygenerated if there exist elements a,, az, ..., a, of A such thatA = (a,, a, ..., a,). An integral domain R is said to satisfy the as-cending chain condition if every strictly increasing chain of idealsICI,c.must be finite in length. Show that an integral domainR satisfies the ascending chain condition if and only if every idealof R is finitely generated.

In this question, we have to show that an integral domain R satisfies ascending chain condition if…

Answered: An ideal A of a commutative ring R with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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17. Given that (I,t.) in an ideal of the ring (R, +,), whow that a) whenever (R, 1,) in commutative with identity, then so is the quotient ring (R/1,+,), b) the ring (R/I,+,) may have divisors of zero, even though (R, +,) does not have any, c) if (R, +,) in a principal ideal ring, then so is the quotient ring (R/I,+,). ive rims with il.untity and kt Nlonete the set of
Right?
7- A ring with identity (R. +..) is commutative if: (a) Ris an integral domain (c) (a + b) - a' + Zab + b' for any a, b eR. (b) Every element in a ring R is idempotent a + Zab + b' for any a, b ER. (d) All answers (a. b,c) are correct.
What is the VaR Controversy?
(5) In commutative ring with identity each proper ideal is contained in prime ideal. От OF
7. Suppose that (R, +,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is not prime ideal then (a) 3a, b E R:a. b eI = a ¢ I and b ¢ 1 (b) Va, b E R: a. b E I = a ¢ I and b ¢ I (c) Va, b e R:a. b EI = a E I or b e I (d) No Choice
6. Suppose that (M, +,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M, then (a) (M, x) = R (b) x R (c) rad M = 0 (d) No Choice
2. Given f is a homomorphism from the ring (R,+, ) onto the ring (R',+', '), prove that a) if (I,+, ) is a maximal ideal of (R,+, ·), then the triple (f(I),+', ') is a maximal ideal of (R',+', '), b) if (I', +', ') is a maximal ideal of (R',+', '), then the triple (f-'(1'),+, ·) is a maximal ideal of (R, +, :).
Both
Let S= { la.beR}. and let :S-R be defined by: :) 1) is a ring a) Homomorphism. b) Isomorphism c) None 2) ker () is generated by b) : ) 3) S/ker (6) = a) b) c) d) None 4) Ker (6) is: a) Prime ideal not maximal b) Maximal ideal not prime c) Both'maximal and prime Setupexe for anything
17. Given that (I, 1.) in an ideal of the ring (R, +,), show that a) whenever (R, 1,) is commutative with identity, then so is the quotiont ring (R/1,+,), b) the ring (R/1,+,) may have divisors of zero, even though (R, +,) docs not have any, c) if (R,+,) is a principal ideal ring, then so is the quotient ring (R/I, +,).

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