TITLE: Is there a relationship between vector spaces and fields/rings/groups?
QUESTION [8 upvotes]: I understand from a comment under Vector Spaces and Groups that every vector space is a group, but not every group is a vector space.
Specifically, I would like to know, can I make a statement like: "All fields are rings, and all rings are groups"?
At this point in my studies, I see various lists of axioms, and I'm trying to see the relationship between them all.
This all because I have a headache, so I went to lie down with a linear algebra book. It's not helping.

REPLY [2 votes]: I would recommend to learn these terms hierarchically: start first with magmas, then magmas with special properties (semigroups), then monoids, then monoid with special properties (groups), then semirings, then rings, then rings with special properties (such as integral domains, factorial rings, fields, etc.). Consider at least one example for every type. Once you know about this, start with modules (over a ring), which includes the special case of vector spaces (which are by definition modules over a field) and of abelian groups (modules over the integers).
You might want to understand the difference between a structure and a property.