**Exercise 9**

**Theorem**:
For any ring , has no subobject classifier.

*Proof*:
Assume it did have a subobject classifier .
Note that the terminal object in is the zero module, which is also initial, so we have , and so there is exactly one choice for the map .
Since is a sub-object classifier, we have for any monic , a unique classifying map such that is the pullback of along .
But, considering as the unique map from the initial object, this is precisely the statement that is the kernel of .
More compactly, this means for *any* submodule of , must be the kernel of its classifying map .
Now consider that the zero module is a subobject of *any* module – thus, for *any* module there must be a map that has trivial kernel – i.e., an injection from into .
Then, in particular, this is true for the -module ( has all small limits).
But, the underlying set of this module has cardinality , a contradiction.