The elements of "C""n" are called "n"-chains and the homomorphisms "d""n" are called the boundary maps or differentials. The chain groups "C""n" may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring "R". The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of "R"-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the category Ab of abelian groups); a celebrated theorem by Barry Mitchell implies the results will generalize to any abelian category. Every chain complex defines two further sequences of abelian groups, the cycles "Z""n" = Ker "d""n" and the boundaries "B""n" = Im "d""n"+1, where Ker "d" and Im "d" denote the kernel and the image of "d". Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as