About the direct sum.

Question

Today I decide to study some topics of Algebra and then faced up with the definition of the direct sum of modules. To let us on the same page, the definition that I'm talking about is

Given a ring $R$ and a family $(M_i)_{i\in I}$ of left $R$-modules, the direct sum of the family $M_i$ to be the set of all sequences $(\alpha_i)$ such that $\alpha_i\in M_i$ and $\alpha_i \neq 0$ for just a finite number of indixes $i\in I$.

I also checked out the definition of the direct product which is basically the same, except the last propertie of $\alpha_i = 0$ for "almost every indixes". My question relies exactly on that propertie. Is there a reason to ask it? Is it demanded just for garantie that the direct sum have a basis? Or exists another readon to justify it. I'm grateful for any help on this subject and I apologize for any error on my english.

Also, this post Why is cofiniteness included in the definition of direct sum of submodules? seems to be the same question, but it doesn't give the answer that I want.

I think you mean "except for the last property", rather than "despite the last property". — Arturo Magidin, Feb 13 '24 at 00:02
"Is there a reason to ask it?" It makes the two things completely different, when there are infinitely many indices... — rschwieb, Feb 13 '24 at 01:42
Btw, modules don't always have bases. So direct sums are not always going to have bases. — David M, Feb 13 '24 at 05:09

score 1 · Answer 1 · answered Feb 12 '24 at 23:59

There is no "reason" to ask for this property -- these are definitions. It is simply the case that sometimes we want to study a module of the form $\bigoplus_i M_i$, and other times we want to study a module of the form $\prod_i M_i$. Both of these kinds of modules exist, and appear very often! So it is useful to have notation to describe them.

score 1 · Accepted Answer · answered Feb 13 '24 at 05:04

Given a family $(M_i)$ of $R$-modules, the direct product $\Pi_i M_i$, together with the family of projection maps $\pi_i \colon \Pi_i M_i \to M_i$, has a very important universal property. Namely, given any module $N$ and module homomorphisms $\phi_i \colon N \to M_i$ for each $i$, there is a unique homomorphism $\phi \colon N \to \Pi_i M_i$ such that for all $i$, $\phi_i = \pi_i \circ \phi$. Moreover, $\Pi_i M_i$ is unique up to a unique isomorphism.

That is a nice property that gets used all the time, so a natural question is whether the picture can be reversed. It turns out that $\bigoplus_i M_i$, together with the injections $\iota_i \colon M_i \to \bigoplus_i M_i$, is exactly what is needed. That is, for any module $N$ and homomorphisms $\psi_i \colon M_i \to N$, there is a unique homomorphism $\psi \colon \bigoplus_i M_i \to N$ such that for all $i$, $\psi_i = \psi \circ \iota_i$. Moreover, $\bigoplus_i M_i$ is also unique up to a unique isomorphism. That provides one illustration of its importance.

About the direct sum.

2 Answers2