]> Commutative Algebra - Direct Sums and Products

Commutative Algebra

Let {M iiI} be a family of R-modules. Define the direct product by iIM i={(x i) iIx iM i} The direct product inherits an R-module structure by defining the operations coordinatewise.

Direct sums of modules iIM 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=M i and M j( ijM i)={0 }.