2. Theorem B: Suppose A, B are submodules of M. PROVE: (A + B)/A≈ B/An B (that is, they are isomorphic as R-modules): Hint: Define a function f B →(A + B)/A and show your f is an onto R-module epimorphism, then use the FHT)
2. Theorem B: Suppose A, B are submodules of M. PROVE: (A + B)/A≈ B/An B (that is, they are isomorphic as R-modules): Hint: Define a function f B →(A + B)/A and show your f is an onto R-module epimorphism, 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. Graduate level.
Note: let R be any ring and M and N R − modules

Transcribed Image Text:2. **Theorem B**: Suppose \( A, B \) are submodules of \( M \). **PROVE**: \( (A + B)/A \cong B/A \cap B \) (that is, they are isomorphic as \( R \)-modules). **Hint**: Define a function \( f : B \rightarrow (A + B)/A \) and show your \( f \) is an onto \( R \)-module epimorphism, then use the FHT (First Isomorphism Theorem).
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