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

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?
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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.

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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.

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