In a linear algebra course, a direct sum of two vector spaces $V, W$ is defined as a vector space $V\oplus W$ where $V \cap W = \{ 0 \}$. And in Group Theory, a direct sum of abelian groups $A_i$ is the group of $\{(a_0, a_1, \ldots) \mid a_i \in A_i \}$ where there is only finitely many non-zero $a_i$. Later on, in Ring Theory, I have learn that the structure of direct sum of ideals process similar properties to vector spaces (the first definition).
The two definitions are
- $V \oplus W = V+W$ for $V \cap W = \{0\}$
- $\bigoplus_{i\in I} A_i = \{ (a_1, \ldots, a_i, \ldots) \mid a_i \in A_i \}$ for which finitely many $a_i$ are non-zero
Both definitions look similar for finite set $I$. But if the first one is extended by induction, then the two definitions do not seem to agree for countably infinite family $I$.
I notice that an Ideal is an Abelian group (along with multiplicative absorbtion). Therefore, I think that there should be a connection between the two direct sum definitions, although I cannot find any explicit reason.
I have been reading a little bit into category, that the direct sum of both cases have certain common universal properties. However, I am seeking for a more specific insight regarding this example, not any two categories.
So, here is my question:
Are both definitions the same? or at least to what extent that they are the same?
If we have a family $A_i, i \in I$ of sets with some binary operation $\circ_i$ and with some dedicated neutral element $0_i$ each, both direct sum $\bigoplus_{i\in I} A_i$ and direct product $\prod_{i \in I} A_i$ are defined set of functions $f: I \to \cup_i A_i$ s.t. $\forall i: f(i) \in A_i$ with operation $(f \circ g)(i) = (f(i))\circ_i (g(i))$.
Difference is that in definition of direct sum there is an additional requirement: that for all but finitely many $i$, we have $f(i) = 0_i$ - ie only finitely many non-zero coefficients.
It's easy to see that if $I$ is finite then both definitions agree, while if $I$ is infinite, direct sum is just subset of direct product.