== Modules in category theory == We aim now to prove that if R {\displaystyle R} is a ring, R {\displaystyle R} -mod is an Abelian category. We do so by verifying that modules have all the properties required for being an Abelian category. Theorem 10.1: The category of modules has kernels. Proof: For R {\displaystyle R} -modules M , N {\displaystyle M,N} and a morphism f : M → N {\displaystyle f:M\to N} we define ker f := { m ∈ M | f ( m ) = 0 } {\displaystyle \ker f:=\{m\in M|f(m)=0\}} . == Sequences of augmented modules == -category-theoretic comment