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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.
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.
Every vector space is over a field. And every field is a vector space over itself.
So for instance $\Bbb R$ is a field. But we can also say that $\Bbb R$ is a vector space over the field $\Bbb R$. Likewise $\Bbb C$ can be considered a vector space over $\Bbb C$ (making it a one-dimensional vector space). Also $\Bbb C$ can be considered a vector space over $\Bbb R$ (making it a two-dimensional vector space).
If we use a ring instead of a field as our scalars, then we get a module rather than a vector space. Then every module is over a ring. And every ring is a module over itself.
Yes, All Fields are rings, and all rings are groups. That said, it is perhaps worthwhile to add a few words of clarification.
Among these three, fields, rings and groups, the groups have the simpler structure. Groups require only one operation among it's members, and it is this operation that needs to satisfy the group axioms. Rings, however, require two operations, sometimes called "addition" and "multiplication", but this is only a convention. These two operations have to cooperate with each other -- this is actually one of the ring axioms. You want to have something like the distributive law that connects addition and multiplication of numbers. This is possibly the reason why the two operations of a ring are often called "addition" and "multiplication".
What might be confusing for beginners, is that with respect to one of these operations - traditionally the "addition", the ring-structure is required to be a group, while the second operation - the "multiplication", it is not required to induce a group, and indeed, one of the most important rings -- the ring of real valued $n\times n$ matrices, has two operations -- addition and multiplication -- and while with respect to the addition operations the matrices form a group, with respect to the multiplication operation they (after we've discarded the zero element) do not form a group, because some matrices do not have a multiplicative inverse. The ring of matrices has a multiplicative identity element, but this is not required by definition. There are rings without a multiplicative identity.
So a ring is a group with an additional structure - that obtained by another operation. A field is a very special kind of ring. Not only do we require that we have two groups with respect to both operations, we also require them to be commutative. This is quite a restriction compared to rings. More precisely, a field is a triple $\{F,+\cdot\}$ such that $\{F,+\}$ is a commutative group and $\{F^*,\cdot\}$ is also a commutative group, where $F^*$ is obtained from $F$ by discarding the additive identity (normally denoted by $0$), such that the operations $+$ and $\cdot$ satisfy the distributive law.
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.
