I know that vector spaces have more structure as they are defined over fields and that modules are defined over rings. But it's hard to think of a situation where a using a ring clearly backfires. Is it just because a ring doesn't have an inverse for the second operation? For a module over Z where addition has an inverse and multiplication doesn't, I could multiply a module element by an integer. Sure I might not be able to reverse it by division. But I could have just not multiplied in the first place. Or I could just keep track of where I came from by storing the previous location if I am using a computer, in that case it would just be more storage space.