I'm just reading Atiyah-Macdonald's commutative algebra text, specifically the section on Operations on Submodules. I just have some questions regarding constructions.
Firstly, suppose $M$ is an $A$-module, and $M' \subset M$ is a submodule. We can then construct the quotient module $M/M'$. I'm wondering if there is any restriction on $M'$? I ask this because when we consider groups, we require that the group be a normal subgroup.
Secondly, if we consider an $A$-module $M$ and let $(M_i)_{i \in I}$ be a family of submodules of $M$. Their sum $\sum_{i \in I} M_i$ is the set of all finite sums $\sum_{i \in I} x_i$, where $x_i \in M_i$ and almost all the $x_i$, that is but a finite number, are zero. I'm lost as to why this condition of zero is here? Is it natural, where does it come from?
Thanks
A module is an abelian group, and so every subgroup (and therefore any submodule) is a normal subgroup. Further, it's easy to show that scalar multiplication is well-defined in the quotient module, and so we don't need any added conditions.
For your second question, that condition (that only finitely many $x_i$ are nonzero) is basically saying it contains all the finite sums of vectors in the respective spaces (so your definition was a bit redundant, you merely specified that the sums have to be finite in two different ways). This condition is necessary because there isn't a good way (for these purposes) to define an infinite sum of elements of a module.
These condition are sufficient to ensure that the sum is the module additively "generated" by the modules which you are summing, i.e. the sum module is the smallest module containing the union of all of the submodules.