Answered: Recall that

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Recall that

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 9E: Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is...

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Question

Recall that a finitely-generated ideal I is written I = (x₁ , x₂ , . . . , x_n) where the x_i ∈ R are
such that every y ∈ I can be written y = r₁x₁ + · · · + r_nx_n for suitable choices of r_i ∈ R.
(a) Let R = Z and I = (4, 6). Show that I is principal and is generated by 2 ∈ Z.
(b) Do the same for I = (6, 9, 15). Find a generator.
(c) Now let R = Z₅[x] and I = (x² + 1, x⁵ + 2). Is I a principal ideal? If so, find a generator.

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