Direct Product - Direct Product of Modules

Direct Product of Modules

The direct product for modules (not to be confused with the tensor product) is very similar to the one defined for groups above, using the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean space Rn, the prototypical example of a real n-dimensional vector space. The direct product of Rm and Rn is Rm + n.

Note that a direct product for a finite index is identical to the direct sum . The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of Category Theory: the direct sum is the coproduct, while the direct product is the product.

For example, consider and, the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in Y. For example, (1,0,0,0,...) is in Y but (1,1,1,1,...) is not. Both of these sequences are in the direct product X; in fact, Y is a proper subset of X (that is, Y⊂X).