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Are Vector Spaces over non-commutative fields ever studied?

If not, why?

If they are, I'd like to learn a bit about them, could you guys recommend me a good book on the subject?

Cheers.

YoTengoUnLCD
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2 Answers2

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The expression "vector space" is usually reserved to the case when the coefficients form a field. When the coefficients only form a ring (such as a skew field), the corresponding structure is called a module.

Joonas Ilmavirta
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    And yes, modules over skew fields (aka division rings) are studied. See e.g. https://en.wikipedia.org/wiki/Division_ring#Relation_to_fields_and_linear_algebra – Robert Israel Jul 07 '15 at 20:49
  • @MattSamuel, I was not aware that modules over skew fields can also be known as vector spaces. All material I've ever come across would have called them only modules. I added a word "usually" to my answer to reflect the fact that it might not always be as I wrote. – Joonas Ilmavirta Jul 08 '15 at 09:02
  • Bourbaki calls a module over a (not necessarily commutative) field a vector space. – Keith Jul 08 '15 at 09:34
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Yes, they are studied, if only because the endomorphism ring of a simple module over a (unital) ring $R$ is a division ring or skew-field, if you prefer. The term “non-commutative field” is rarely used.

Wedderburn's theorem says that a semisimple artinian ring is the product of (full) matrix rings over division rings, generally non commutative.

However, the structure of vector spaces over a division ring $D$ is quite easy: $D$ is a semisimple ring and has a unique simple module, up to isomorphisms (the same $D$ as a vector space over itself). So any module over a division ring $D$ is just a direct sum of copies of $D$, just like for vector spaces over fields.

The theorems about linear independence and bases extend to this setting and also the duality for finite dimensional vector spaces.

egreg
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