Structures where the injectivity of morphisms is forced

Question

Can someone give me some examples of mathematical structures where the associated morphisms are forced to be injective (e.g. fields)?

Thanks

Answer 1

Schur's lemma is a more broad result in this direction. The rule of thumb is that you know for a given category (of course, when applicable) what the kernel is going to look like, so if your object doesn't have nontrivial subobjects that look like that (like a field, or simple module) nonzero morphisms are going to be injective.

Answer 2

In any algebraic setting where you have a category $C$ with a good notion of kernel, and where a morphism is mono (or becomes injective after applying the forgetful functor to $Set$ in the concrete case) if, and only if, its kernel is $0$ you can produce a category where all non-zero morphisms are monos as follows. The category $D$ in the full subcategory of $C$ spanned by the simple objects, where an object $X$ in $C$ is simple if it has no non-trivial normal subobjects (a normal subobject is a subobject which is a kernel of some morphism).

In particular, the category of simple groups and group homomorphisms is an example, as well as many other similar situations.