Let M=⨁i∈ISiM=\bigoplus_{i\in I} S_{i} be a semisimple module, with SiS_{i} simple submodules ∀i∈I\forall i \in I. We know that if N⊆MN\subseteq M is a submodule (N≠{0}N\neq \{0\}) then it is a direct sum of some subset, i.e. N≅⨁i∈JSiN\cong\bigoplus_{i\in J} S_{i} with J⊆IJ\subseteq I. But if NN is a proper submodule of MM, then JJ is a proper subset of II?

For example, can we have S1⊕⋯⊕Sn≅S1⊕⋯⊕SmS_{1}\oplus \cdots\oplus S_{n}\cong S_{1}\oplus\cdots\oplus S_{m} with n>mn\gt m and SiS_{i} simple submodules ∀i\forall i?

=================

=================

=================