equalizer of modules
1. Proposition
Let 
be a ring, 
be modules and 
a module-homomorphism.
Then the equalizer is the submodule
2. Proof
2.1. kernel construction
2.2. submodule
follows from
- Menge der Modulhomomorphismen als abelsche Gruppe, hence
is a module homomorphism - Kernel of a module homomorphism as submodule
2.3. universal property
Suppose there exists a module homomorphism 
making this a fork
Then we define
by restriction of a module homomorphism to the image as module homomorphism it remains to show, that this map is welldefined:
For 
, it follows that
hence
or 