Direct Sum of Modules - Construction For An Arbitrary Family of Modules

Construction For An Arbitrary Family of Modules

One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of two modules. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows (Bourbaki 1989, §II.1.6).

Let R be a ring, and {M_i : i ∈ I} a family of left R-modules indexed by the set I. The direct sum of {M_i} is then defined to be the set of all sequences where and for cofinitely many indices i. (The direct product is analogous but the indices do not need to cofinitely vanish.)

It can also be defined as functions α from I to the disjoint union of the modules M_i such that α(i) ∈ M_i for all i ∈ I and α(i) = 0 for cofinitely many indices i. These functions can equivalently be regarded as finitely supported sections of the fiber bundle over the index set I, with the fiber over being .

This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences (or functions) α and β can be added by writing for all i (note that this is again zero for all but finitely many indices), and such a function can be multiplied with an element r from R by defining for all i. In this way, the direct sum becomes a left R-module, and it is denoted

It is customary to write the sequence as a sum . Sometimes a primed summation is used to indicate that cofinitely many of the terms are zero.