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I know this is a question asked many times, but I'm looking to understand a particular thing which is not really mentioned elsewhere.

Let $M,N$ be left $R$-modules, and the definitions for their direct sum and direct product I am familiar with are:

$M\oplus N = \{m+n : m\in M, n\in N\}$, where the expression $m+n$ is unique for each element in the direct sum.

$M\times N =\{ (m,n) : m\in M, n\in N\}$, with operations performed componentwise.

For finitely many modules, these are isomorphic, by $\sum_i^k m_i \leftrightarrow (m_0,\ldots, m_k)$.

Now the difference is when there is an infinite collection of modules: The direct sum is a submodule of the direct product, such that it's elements (written as a tuple, I guess) have only finitely many non-zero components.

Where does this restriction come from? We are working with formal sums in the direct sum, so we couldn't be concerned with 'convergence' or anything, right? I just can't see (nor find an explanation) why we can't have infinite formal sums. Does something about the categorical definition break down? Though again, not sure what stops us having suitable injections.. (Category theoretic definitions are rather new to me - in fact, that I don't see what could make the diagram go wrong for infinite sums is one reason I have to ask this.)

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    $\begingroup$ The very definition of (rings and) $R$-modules is restricted to finite addition only, that is the reason for the difference in the infinite case. $\endgroup$
    – Berci
    Sep 7, 2015 at 19:41

1 Answer 1

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If you allow infinite sums, you again obtain a direct product. The two different objects indeed correspond to different (dual) category theoretic notions: A direct product of modules is a product in the module category, while a direct sum is a coproduct. They satisfy the following respective universal properties, that characterize them:

Let $(M_i)_{i \in I}$ be a family of (left) $R$-modules.

  1. A module $P$, together with a family of homomorphisms $(p_i \colon P \to M_i)_{i \in I}$, is a direct product (product) of the family $(M_i)_{i\in I}$, if for every module $X$ and every family of homomorphisms $(f_i \colon X \to M_i)$, there exists a unique homomorphism $f \colon X \to P$ such that $f_i = p_i \circ f$ (for all $i$).

  2. A module $C$, together with a family of homomorphisms $(e_i \colon M_i \to C)_{i \in I}$ is a direct sum (coproduct) of the family $(M_i)_{i \in I}$, if for every module $X$ and every family of homomorphisms $(f_i\colon M_i \to X)_{i \in I}$, there exists a unique homomorphism $f \colon C \to X$ such that $f_i = f \circ e_i$ (for all $i$).

If you draw the usual diagrams for these (which you have probably seen), you see that one is obtained from the other by inverting all the arrows. In particular, a direct product comes with a family of projections onto the original family of modules. Conversely, a direct sum comes with a family of embeddings of the original family of modules. They are each universal in their respective sense in that every other such diagram "factors" through them. This is what makes them important from the perspective of category theory. They are also unique up to unique homomorphism by abstract nonsense, so if you find any construction which satisfies (1), you have found "the" direct product; similarly for direct sums.

If you try to verify the second property for the direct product (infinite tuples), you will hit a problem: You'd take for the embeddings the obvious ones into your tuples, i.e., $M_i$ is simply mapped into the $i$-th component of $\prod_{i \in I} M_i$. But now, how do you define $f \colon \prod_{i \in I} M_i \to X$? For the direct sum, you simply take $m_{i_1} + \cdots + m_{i_n} \mapsto f_{i_1}(m_{i_1}) + \cdots + f_{i_n}(m_{i_n})$. But this only works using that the elements are finitely supported, since the last sum is indeed an "actual" sum in $X$.

That the product and coproduct coincides for a finite family of modules is very useful, but perhaps obscures things a little bit in the beginning, because it may be hard to get a intuition for the difference. It may help to consider another category. E.g. in the category of sets, products are given by Cartesian products, while coproducts are disjoint unions of sets. These are quite different, even for finite families.

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