So my Linear algebra final is coming up, and I am very confused about the property of direct summands and how it relates to pure submodules.
Now if we have a submodule $N$ of $M$ is pure when for any $y\in N$ and $a\in D$, where $D$ is the domain, if there exist $x \in M$ with $ax = y$ then there exists $z\in N$ with $az = y$.
So Now why is it trivial that if $N$ is a direct summand, then $N$ is pure in $M$ ?
Is it because a direct summand is closed under addition. I understand its supposed to be really easy to understand, but I cannot get over the hump of understanding this aspect, and it's important to understand for the following question.
For $x \in M$, let $x + N$ denote the coset which is the image of $x$ in the module $M/N$. If $N$ is a pure submodule of $M$, and $ann(x + N)$ is a principal ideal, $d$, of Domain $D$, prove that there exists $x' \in M$ such that $x + N = x' + N$ and $ann(x')$ is the same ideal as $ann(x+N)$
Any help in understanding this concept and how it implies to the second question would be amazing! Thanks in advance!
EDIT:: So I think I got the first part we have $M = N \oplus K$ then for $x \in M$, $x = n + k$ where $n\in N$ and $k\in K$. so $ax = an + ak$ then $ax = an$ as $ak \in N \cap K = {0}$. Thus $N$ is pure in $M$. Still trying to figure out Part 2.
It's because you have a projection onto the direct summand, which projects a solution in M (to any system of linear equations, in fact) to a solution in N. (Note, homogeneous linear equations don't change under linear maps.)