Direct Sums and Products

Let \(\{M_i | i\in I\}\) be a family of \(R\)-modules. Define the direct product by

\[ \prod_{i\in I} M_i = \{ (x_i)_{i\in I} | x_i \in M_i \} \]

The direct product inherits an \(R\)-module structure by defining the operations coordinatewise.

Direct sums of modules \(\oplus_{i\in I} M_i\) are defined in the same way except only finitely many of the coordinates of the tuples are nonzero. Thus if \(I\) is finite direct sums and products are the same.

We say \(M\) is the internal direct sum of the family if \(M = \sum M_i\) and \(M_j \cap (\sum_{i \ne j} M_i) = \{ 0 \}\).

Ben Lynn 💡