# Structures where the injectivity of morphisms is forced

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

Thanks

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There is no notion of injective morphism in a category. Do you mean monomorphism? Or do you refer to concrete categories and define injectivity via the forgetful functor? –  Martin Brandenburg Apr 14 '13 at 8:44
Yes, concrete categories. –  Henry Apr 14 '13 at 8:57

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).