Easy Exercise on Modules

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Joined: Tue Nov 10, 2015 8:25 pm

Easy Exercise on Modules

#1

Post by Tsakanikas Nickos »

Let \( V_{1},V_{2} \) be two modules and let \( U_{i},W_{i} \) be submodules of \( V_{i} \). Show that:
  1. \( \left( U_{1} \bigoplus U_{2} \right) \cap \left( W_{1} \bigoplus W_{2} \right) = \left( U_{1} \cap W_{1} \right) \bigoplus \left( U_{2} \cap W_{2} \right) \)
  2. \( \left( U_{1} \bigoplus U_{2} \right) + \left( W_{1} \bigoplus W_{2} \right) = \left( U_{1} + W_{1} \right) \bigoplus \left( U_{2} + W_{2} \right) \)

Does the following relation hold?
\[ \left( U_{1} \oplus U_{2} \right) \cap W = \left( U_{1} \cap W \right) \bigoplus \left( U_{2} \cap W \right) \]
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