Direct products vs. direct sums vs. sums

#module_theory

Suppose ${N_{s} ∣ s \in S}$ is a family of submodules of a fixed $R$ -module $M$ . We have four constructions available to create a new module:

Constructions as submodules

Let's first consider ${N_{s}}_{s \in S}$ as a family of objects in the category (whose objects are the submodules of ).

The intersection of this family of submodules

In this category, the product of this family is called their intersection and is denoted $⋂_{s \in S} N_{s}$ . As a set, its elements are those elements contained in every submodule ; i.e., it is the set-theoretic intersection of the sets of elements of all $N_{s}$ . It is a submodule of $M$ , the largest submodule of $M$ contained within every submodule $N_{s}$ in the original family.

The sum of this family of submodules

Still considered as a family of objects in $M$ , the coproduct of the family is their sum and is denoted $\sum_{s \in S} N_{s}$ . As a set, its elements are all finite $R$ -linear sums of the form $\sum_{s \in S} r_{s} n_{s}$ , where each $n_{s} \in N_{s}$ and $r_{s} \in R$ (all but finitely many of which are zero). It is a submodule of $M$ , the smallest submodule of $M$ that contains every submodule $N_{s}$ in the original family. In this module, different sums may represent the same element.

Constructions as modules

Now let's consider ${N_{s}}_{s \in S}$ as a family of objects in the category $R - Mod$ .

The direct product of this family as modules

In this category, the product of this family is called the (direct) product and is denoted $\prod_{s \in S} N_{s}$ . As a set, its elements consist of maps $f$ from $S$ such that $f (s) \in N_{s}$ for every $s \in S$ . When $S$ is finite, its elements can equivalently be viewed as $S$ -tuples of elements $(n_{s})_{s \in S}$ with $n_{s} \in N_{s}$ for each $s \in S$ . It is not a submodule of $M$ , nor does it literally contain any of the original submodules ; however, there are surjective morphisms from $\prod_{s \in S} N_{s}$ to each $N_{s}$ , and in the finite case there are injective morphisms from each $N_{s}$ into $\prod_{s \in S} N_{s}$ . This also means that we have an injective morphism $⨁_{s \in S} N_{s} \to \prod_{s \in S} N_{s}$ . In the case of a finite set $S$ , this is an isomorphism.

The direct sum of this family as modules

In this category, the coproduct of this family is called their direct sum and is denoted $⨁_{s \in S} N_{s}$ . As a set, its elements are sometimes described as all formal finite $R$ -linear sums of the form $\sum_{s \in S} r_{s} n_{s}$ , where each $n_{s} \in N_{s}$ and $r_{s} \in R$ (all but finitely many of which are zero). As mentioned previously, however, we should really view the elements of $⨁_{s \in S} N_{s}$ as functions $f : S \to Z$ where $f (s) \in N_{s}$ , all but finitely many zero (and where $Z$ is any set containing all of the elements of every $N_{s}$ ).

In any case, it is not a submodule of $M$ , nor does it literally contain any of the modules from the original family; however, it does contain a copy of each of those submodules. In this module, different "formal sums" always represent different elements. There are no relations or simplifications, beyond that of combining or simplifying coefficients.

Relationships between these constructions

As mentioned previously, whenever $S$ is finite there is a natural isomorphism between the direct product over $S$ and the direct sum over $S$ . When $S$ is infinite, from our constructions we can see that there is an injective (but not surjective) module morphism $⨁_{s \in S} M_{s} \to \prod_{s \in S} M_{s}$ .

There is also a special case in which the sum of a family of submodules is isomorphic to the direct sum of the family of submodules (viewed as modules).^[1] We'll state the finite result for simplicity.

When a sum is a direct sum

Suppose $N_{1}, \dots, N_{k}$ is a family of submodules of an $R$ -module $M$ . There is an $R$ -module isomorphism

N_{1} \oplus \dots \oplus N_{k} ≃ N_{1} + \dots + N_{k}

exactly when $N_{j} \cap (N_{1} + \dots + \hat{N_{j}} + \dots + N_{k}) = (0)$ for every $j$ . In this case, the map is given simply by sending each formal sum $n_{1} + \dots + n_{k}$ to the corresponding (actual) sum in $M$ .

A word on language

The terms "Cartesian product" (of sets) and "direct product" (of modules, groups, etc.) is a bit antiquated. Each of these types of objects satisfies the "same" universal property from the point of view of category theory, which is simply that of a product. To make matters unnecessarily confusing, categorical products are special cases of a more general notion of limit, which was sometimes called "inverse limit" or "projective limit". This was to distinguish them from the dual of notion of colimit, which was once called "direct limit" or "inductive limit". So a direct product is an example of an inverse limit (not a direct limit), while a direct sum is an example of a colimit (or direct limit). Oof. Let's be kind to ourselves and stick simply with "product" and "coproduct".

Suggested next notes

Natural transformations
Free modules

In the finite case, this is also isomorphic to the direct product of those submodules viewed as modules. ↩︎