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Tiffy
Tiffy
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Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.

Given two free o-submodules $L_1,L_2$ in $V$ such that $K \cdot L_i \subsetneqq V$ for $i=1,2$, does then also $K \cdot (L_1+L_2) \subsetneqq V$$L_1+L_2 \subsetneqq V$ hold? (Here L_1+L_2 denotes the smallest $o$-module inside $V$ containing $L_1$ and $L_2$)

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.

Given two free o-submodules $L_1,L_2$ in $V$ such that $K \cdot L_i \subsetneqq V$ for $i=1,2$, does then also $K \cdot (L_1+L_2) \subsetneqq V$ hold? (Here L_1+L_2 denotes the smallest $o$-module inside $V$ containing $L_1$ and $L_2$)

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.

Given two free o-submodules $L_1,L_2$ in $V$, does $L_1+L_2 \subsetneqq V$ hold? (Here L_1+L_2 denotes the smallest $o$-module inside $V$ containing $L_1$ and $L_2$)

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Tiffy
Tiffy
  • 107
  • 5

Sum of two free o-submodules in a vector space over a local field

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.

Given two free o-submodules $L_1,L_2$ in $V$ such that $K \cdot L_i \subsetneqq V$ for $i=1,2$, does then also $K \cdot (L_1+L_2) \subsetneqq V$ hold? (Here L_1+L_2 denotes the smallest $o$-module inside $V$ containing $L_1$ and $L_2$)