Let R be a commutative ring and let I and J be ideals in R. Define the sets I + J and IJ as follows: I+J= {i+ji EI, jЄ J} and IJ = {i1j1 + i2j2 + +ikjk 11, 12, ‚ ik Є Ï‚ ι‚ Ĵ2, . . ., Ìk Є J}. (a) Prove that In J, I + J, and IJ are ideals of R. (b) If B – FX1 I — ( f (X)) and I — (a(X)) find explicitly LO T + I and II
Let R be a commutative ring and let I and J be ideals in R. Define the sets I + J and IJ as follows: I+J= {i+ji EI, jЄ J} and IJ = {i1j1 + i2j2 + +ikjk 11, 12, ‚ ik Є Ï‚ ι‚ Ĵ2, . . ., Ìk Є J}. (a) Prove that In J, I + J, and IJ are ideals of R. (b) If B – FX1 I — ( f (X)) and I — (a(X)) find explicitly LO T + I and II
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 be a commutative ring and let I and J be ideals in R. Define the sets I + J
and IJ as follows:
and
I + J = {i + j | i Є I, jЄ J}
IJ := {i1j1 + i2j2 + + İkİk | i1, i2, . . ., ik Є I, ̹, Ì2, . . ., Ìk Є J}.
...
(a) Prove that In J, I + J, and IJ are ideals of R.
(b) If R = F[X], I
=
(f(X)), and J = (g(X)), find explicitly In J, I + J, and IJ,
i.e., for each of them find a generator.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf58e778-209d-4051-b758-0c57e3f8187c%2Fcb4fcbd0-3141-459d-a19d-fef92b14e957%2Fmex43mb_processed.png&w=3840&q=75)
Transcribed Image Text:Let R be a commutative ring and let I and J be ideals in R. Define the sets I + J
and IJ as follows:
and
I + J = {i + j | i Є I, jЄ J}
IJ := {i1j1 + i2j2 + + İkİk | i1, i2, . . ., ik Є I, ̹, Ì2, . . ., Ìk Є J}.
...
(a) Prove that In J, I + J, and IJ are ideals of R.
(b) If R = F[X], I
=
(f(X)), and J = (g(X)), find explicitly In J, I + J, and IJ,
i.e., for each of them find a generator.
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