Let R = Z ⊕ Z ⊕ ...(the collection of all sequences of integersunder componentwise addition and multiplication). Show that Rhas ideals I1, I2, I3, . . . with the property that I1 ⊂ I2⊂ I3⊂,....(Thus R does not have the ascending chain condition.)

Answered: Let R = Z ⊕ Z ⊕ ...(the collection of…

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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Let R = Z ⊕ Z ⊕ ...(the collection of all sequences of integers
under componentwise addition and multiplication). Show that R
has ideals I₁, I₂, I₃, . . . with the property that I₁ ⊂ I₂⊂ I₃⊂,....
(Thus R does not have the ascending chain condition.)

Expert Solution

Step 1

It is given that, $R = ℤ \oplus ℤ \oplus . . .$ be the collection of all sequences of integers under componentwise addition and multiplication.

We have to show that, R has ideals $I_{1}, I_{2}, I_{3}, . . .$ with the property $I_{1} \subset I_{2} \subset I_{3} \subset, . . .$

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