Cohesive picture of groups, rings, fields, modules and vector spaces.

Question

If I understand my algebra correctly every field is a ring and every ring is a group, so when we define modules over rings and vector spaces over fields, we then have that every vector space is a module?

A linear algebra is defined in Hoffman's book as follows.

Let $F$ be a field. A linear algebra over the field $F$ is a vector space $\mathcal{A}$ over $F$ with an additional operation called multiplication of vectors which assosciates with each pair of vectors $\alpha, \beta \in \mathcal{A}$ a vector $\alpha \beta \in \mathcal{A}$ called the product in such a way that,

• Multiplication is associative: $\alpha (\beta \gamma) = (\alpha \beta) \gamma$
• Multiplication is distributive with respect to addition: $\alpha(\beta + \gamma) = \alpha \beta + \alpha \gamma$ and $(\alpha + \beta)\gamma = \alpha \gamma + \beta \gamma$
• For each scalar c in $F$: $c(\alpha \beta) = (c \alpha)\beta = \alpha(c \beta)$

Is there a more recent term for what Hoffman means by a linear algebra and how does it fit into the whole group-ring-field hierarchy? Lastly, if you define a vector space over a field and a module over a ring, what is defined over a group in this same way?

To reiterate, I have three questions,

• Is every vector space a module?
• What relationship does a linear algebra to have to vector spaces and modules?
• Is there an algebraic structure defined over groups the same way that vector spaces are defined over fields and modules are defined over rings?

Answer 1

1. Yes: if $F$ is a field, then "module over $F$" is the same thing as "vector space over $F$". Said another way: vector spaces are modules in which the ring of scalars is a field.

2. A "linear algebra" (or more generally, an "$F$-algebra") is both a ring and an $F$-vector space, in such a way that the ring multiplication is compatible with the $F$-vector space structure.

3. You have "group actions" if all you have is a set and a map $G\times X\to X$ which is compatible with the operations of $G$. If $X$ has an algebraic structure of its own, e.g., if $X$ is an abelian group, then we talk about $G$-modules (which amounts to having a group homomorphism $G\to\mathrm{Aut}(X)$, where "Aut" are the appropriate structure automorphisms).

All of these can be further generalized to the concept of "general/universal algebra" (fields are not universal algebras, but they can be obtained by weakening the conditions to obtain 'partial algebras'). A great introduction to that is George Bergman's An Invitation to General Algebra and Universal Constructions.

Answer 2

Every vector space is a module. The scalars in a vector space come from a field, the ones in a module from a ring.

The current term is simply algebra instead of the older linear algebra.

An algebra is a ring that is also a vector space and its operations are compatible. You can also have algebras over rings, in which case you have a ring that is also a module. Do not mix the field or ring of scalars with the ring of the algebra. There really are two rings here.

The corresponding notion for groups is a group action.

Answer 3

See Arturo's excellent answer above (below?). To this, I wish to add the following:

The definition of an algebra given by Hoffman and clarified by Arturo above is actually that of an associative algebra. Many of the algebras we encounter in classroom mathematics are associative,but it should be noted there are several important types of nonassociative algebras that appear in both algebra and particle physics; algebras where Hoffman's first axiom of the definition fails. The most famous examples are probably the octonions, alternative algebras and the Jordan algebras. And believe it or not,one of the most famous examples is usually studied in high school calculus and is almost never described this way: Three dimensional Euclidean space is a nonassociative algebra under ordinary vector addition and the cross product.In fact, it is both noncommutative and nonassociative under the cross product. (Try and prove it-good exercise for a beginning algebra student!) For much more on nonassociative algebras, the standard text is Richard Schafer's An Introduction To Nonassociative Algebras. A more recent and comprehensive source is Kevin McCrimmon's A Taste Of Jordan Algebras .