When does the short exact sequence of Module and submodule not split

abstract-algebracommutative-algebraexact-sequencehomological-algebramodules

Consider a commutative Ring $R$ and a module $M$ over $R$. Now let $N \subset M$ be a submodule. Then we have a canonical short exact sequence:
$$0 \rightarrow N \xrightarrow{i} M \xrightarrow{p}M/N \rightarrow 0$$

The $i$ is the inclusion and $p$ is the projection onto the quotient module. Up until now, I was under the impression that this sequence always splits by the splitting Lemma. The way I thought about it was, that the map
$$ r: M/N \rightarrow M\; ; \; m+N \mapsto m$$
defines a right-inverse to $p$, i.e. $p \circ r = id_{M/N}$. Since there are multiple $m$, mapping to the same element in $M/N$, I would choose a member of each class, sucht that the map becomes a homomorphism.

But I recently read that it only splits, if $N$ is a direct summand of $M$, which means that there is another submodule $N'$ such that $N \oplus N' = M$. But the existence of $N'$ somehow makes the splitting "obsolete", since this is exactly the definition of a split sequence. In other words, this is an if and only if statement. So $N$ is a direct summand iff the above SES splits. But this doesn't give me any more information on when this sequence splits… So my questions are the following:

  1. Am I allowed to define $r$ in this way, and if yes, why is the map $r$ not a right-inverse to $p$?
  2. Are there ways to check, if $N$ is a direct summand, without trying to split the above SES?
  3. Can someone provide a counterexample of $N$ and $M$ such that the sequence does indeed not split?

Edit: I guess I have to clarify my question:
In the left map, I really want the canonical inclusion of $N$ into $M$ as a submodule. For example, I won't allow the sequence $0 \rightarrow \mathbb{Z} \xrightarrow{\cdot 2} \mathbb{Z} \rightarrow \mathbb{Z}_2 \rightarrow 0$. In this case, the left map would be $id$ and the right one $0$.

Best Answer

  1. Why is your $r$ well-defined? To make it easier for you, think about an explicit example: Is $$\mathbb{Z}/42 \mathbb{Z} \to \mathbb{Z}, \ n + 42 \mathbb{Z} \mapsto n$$ well-defined? (What are all maps $\mathbb{Z}/42 \mathbb{Z} \to \mathbb{Z}$?)
  2. Depending on the situation, there are many. For example, you can check whether the sequence is left-split. If you know that $N$ is an injective module or that $M/N$ is projective, then you also immediately know that the sequence splits. All of this is subsumed by the computation of $\operatorname{Ext}^1$-groups. Note however that being a direct summand is not the same thing as splitting, but you're probably interested in the splitting question for now.
  3. Which examples have you looked at? There are plentiful of such examples. Look back at the example I suggest in 1 and play with it.
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