-se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT
-se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT
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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Abstract algebra 2 . Higher level
Note: let R be any ring and M and N R − modules
![## Theorem C
**Statement:**
Suppose \( A, B \) are submodules of \( M \) with \( A \subseteq B \). **Prove:**
\[
M/A/B/A \cong M/B
\]
This means they are isomorphic as \( R \)-modules.
**Hint:**
Define a function \( g : M/A \rightarrow M/B \) and demonstrate that your function is a well-defined, \( R \)-module epimorphism. Then, apply the First Isomorphism Theorem (FHT).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F30beaf0c-9512-47a8-9b9b-114055cfca4a%2F083jqen_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Theorem C
**Statement:**
Suppose \( A, B \) are submodules of \( M \) with \( A \subseteq B \). **Prove:**
\[
M/A/B/A \cong M/B
\]
This means they are isomorphic as \( R \)-modules.
**Hint:**
Define a function \( g : M/A \rightarrow M/B \) and demonstrate that your function is a well-defined, \( R \)-module epimorphism. Then, apply the First Isomorphism Theorem (FHT).
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