If a chain is exact, it should be called an exact chain, but it's not. Instead, it is called an exact sequence. I guess the word "exact" tells you it's a chain, so you know what is going on. I know, I don't like it either.
If an exact sequence begins 0 → A → B, the image of 0 is 0, which becomes the kernel of A, hence A embeds in B.
If an exact sequence ends B → C → 0, C is the kernel of the second homomorphism, and the image of the first, hence B maps onto C.
If A is a submodule of B, and C equals B/A, the following is an exact sequence; in fact it is called a short exact sequence.
0 → A → B → C → 0
If A and C are finitely generated, these generators combine to produce the module B, hence B is finitely generated. Conversely, assume B is finitely generated and R is noetherian. This makes B a noetherian module, whence A and C are also noetherian. Since Z is noetherian, B is a finitely generated abelian group iff A and C are finitely generated abelian groups.