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.
Best Answer
That is all correct. A field satisfies all ring axioms plus some extra axioms, so a field is a ring. A ring is an Abelian group plus some more axioms, so each ring is a group.
A vector space is also an Abelian group with some extra axioms relating it to a field. The field is an indispensable part of the definition of the vector space.
If you define a vector space to be an Abelian group V which has multiplication defined with a field (or division ring ) $V\times F\to V$ satisfying some axioms, then you can replace F with another ring and do something similar, except that V is called a module over the ring rather than a vector space. In other words, rings and modules are a generalization of fields and vector spaces.