Just finished a semester in abstract algebra. We learned about free abelian groups on finite generators and the Smith Normal Form algorithm to find the elementary divisors which describe said free abelian group modded out by a subgroup.
Seeing the Smith Normal Form proof and seeing it in action had me wondering what the idea of an eigenvalue and eigen vector is when the "linear transformation" looking thing is in a ring and not field.
Are there theorems, notions, proofs, etc. that help connect the way we find $F(n)/H$ with the notion of vectors centered at the origin which don't change up to magnitude? Oh and $F(n)$ is the free abelian group on $n$ generator and $H \leq F(n)$ is the factor group for $H$ is a subgroup of $F(n)$