Is there a classification theorem for "finitely generated abelian groups with a distinguished element"? If it helps, you can restrict this to the cases where the order of each element divides the order of the distinguished element.


My idea would be to try to use this for a classification theorem for rings with a finitely generated additive group, where the distinguished element is 1.



slight extension to classification of finitely generated abelian groups
You say that we may assume that the order of every element divides the order of the distinguished element. That's the easy case. If the distinguished element generates a finite cyclic group $C$ of order $n$ and every element has order dividing $n$, then in fact $C$ is a direct summand. ("For the ring $\mathbb Z/n$ the free module of rank one is injective.")


