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I know that a commutative ring with an additional scalar multiplication on it is called an associative algebra. If the ring also has a 1 it is called a unital algebra. What would you call a field with an additional scalar multiplication(from a different field) on it?

Wintermute
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  • Good question. As I'm sure you're well aware, take a field $K$, then the rational function field $K(x)$ is an example of what you're talking about, perhaps the canonical example. – goblin GONE Jul 03 '14 at 14:44
  • It's just a field with a subfield. – Alex B. Jul 03 '14 at 14:45
  • Just like R-algebras are rings containing a central image of R. – Bill Dubuque Jul 03 '14 at 14:59
  • @AlexB. I'll have to think about that, it's not quite clear to me. One example I'm thinking about is the collection of positive definite symmetric circulant matrices. I'm not sure how this example is a field with a subfield. – Wintermute Jul 03 '14 at 15:43
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    Consider the orbit of the unit element under scalar multiplication. That will give you a copy of the scalar field inside the field (use the fact that a field has no non-zero proper ideals). In your example, the subfield consists of scalar matrices. Conversely, whenever you have a field with a subfield, the big field is clearly an algebra under the subfield, just using multiplication in the big field. – Alex B. Jul 03 '14 at 15:54
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    @AlexB. Thanks, this is a nice way to think about it. Further research also seems to suggest it is an F-algebra (which just seems to be a special type of R-algebra). – Wintermute Jul 03 '14 at 15:58

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An associative algebra over a commutative ring is just an associative ring $A$ which has some scalar multiplication from a commutative ring $R$ which satisfies some axioms.

If both $A$ and $R$ are fields, I'd still call $A$ an $R$-algebra. You might improve by calling it an $R$ division algebra.

rschwieb
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  • I'd call $A$ a field extension of $R$. – lhf Jul 03 '14 at 15:55
  • I found a document that refers to this structure as a F-algebra. http://darkwing.uoregon.edu/~brundan/math647fall04/ch4.pdf – Wintermute Jul 03 '14 at 15:56
  • @mtiano It kind of looks like you're attaching special significance to the letters $R$ and $F$. If you replaced the letter $R$ above with any other commutative ring (such as a field $F$, a field $k$ or an integral domain $D$, you'd have an $F$-algebra, a $k$-algebra, or a $D$-algebra. – rschwieb Jul 03 '14 at 17:24
  • Dear @lhf : It's true, but I felt like this is most in the spirit of the OP's original post, relating them to $R$-algebras. Field extensions are structures I suppose, but they aren't really defined as more than one field containing another, so the algebra relationship (while there) is silent. – rschwieb Jul 03 '14 at 17:38
  • @rschwieb, you're quite right. – lhf Jul 03 '14 at 17:56
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If $k$ is a commutative ring, then a commutative $k$-algebra $A$ is the same as a homomorphism of rings $k \to A$. If $k,A$ are fields, then $k \to A$ is injective. Therefore this is just a field extension.

Martin Brandenburg
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