7. (a) Letbe an increasing sequence of ideals in a ring R. Prove that the union of all these I, is anideal.(b) Give an example of two ideals I and J in a ring R such that IUJ is not an ideal.

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Answered: 7. (a) Let be an increasing sequence of…

7. (a) Let be an increasing sequence of ideals in a ring R. Prove that the union of all these 1, is an ideal. (b) Give an example of two ideals I and J in a ring R such that IUJ is not an ideal.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 30E: a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R....

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7. (a) Let
be an increasing sequence of ideals in a ring R. Prove that the union of all these I, is an
ideal.
(b) Give an example of two ideals I and J in a ring R such that IUJ is not an ideal.

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