Associated primes turns short exact sequences to sub-unions

Statement

Suppose $A$ is a commutative unital ring. Consider a short exact sequence of modules:

$0\to M\to N\to L\to 0$

Then we have:

$Ass_{A}(N)\subset Ass_{A}(M)\cup Ass_{A}(L)$

where $Ass_{A}$ denotes the set of associated primes.

Proof

Proof outline

The key nontrivial ingredient, where we actually use primeness, is the following fact:

• Principal ideal is isomorphic to integral domain as a module: Any principal ideal in an integral domain is isomorphic to the integral domain as a module. Another way of putting this is that any submodule of an integral domain, contains a submodule isomorphic to the whole domain (as a module).

Hands-on proof

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