Question: > 
> Suppose you flip a fair coin three times,
>  find three events $A$, $B$, and $C$, such that no two of them are
>  independent, but $P(A \cap B \cap C) = P(A)P(B)P(C)$
> 
> 
> 

I am given the question above. From what I can tell, this is a trick question since:

$P(A \cap B \cap C) = P(A)P(B|A)P(C|B,A) = P(A)P(B)P(C)$

then 

$P(B|A)$ must equal $P(B)$ (and the same for $P(C|B,A) = P(C)$), which means 

$P(A \cap B) = P(A)P(B)$, that is, they are independent, which goes against the premise.

Am I missing something here or is this correct? (I feel like I'm missing something)

EDIT: If someone could answer this visually (using a venn diagram or something) I'd really appreciate it

Answer: 
> 
> Specifically, I would like to know, can I make a statement like: "All fields are rings, and all rings are groups"?
> 
> 
> 

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.
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.*