# [SOLVED] Vector spaces over division rings.

#### ENRIQUESTEFANINI

Hi:
I have the following definition: let V be a vector space over
a division ring D. A mapping a of V into V is called a linear trans-
formation of V if it has the followiwng two properties:
(x+y)a = xa+ya for x,y $$\displaystyle \in$$ V,
(x$$\displaystyle \alpha$$)a = (xa)$$\displaystyle \alpha$$ for x $$\displaystyle \in$$ V, $$\displaystyle \alpha$$ D

And here I find an odd thing. If a is the mapping multiplication by
a scalar (that is, by an element of D), then a is not in general a linear trans-
formation of V according to the definition, because D needs not be
commutative. Any hint will be welcome.

#### HallsofIvy

MHF Helper
First, in English, at least, a set with operations of addition and "scalar multiplication" with the scalars from a division ring rather than a field is usually called a "module", not a "vector space".

Second, If you define scalar multiplication by "$$\displaystyle x\alpha$$" for $$\displaystyle \alpha$$ in D, then you must either define $$\displaystyle \alpha x= x\alpha$$ or leave $$\displaystyle \alpha x\(\displaystyle undefined. The fact that multiplication of scalars is not commutative has nothing to do with the relationship between \(\displaystyle x\alpha$$ and $$\displaystyle \alpha x$$.\)\)

#### roninpro

First, in English, at least, a set with operations of addition and "scalar multiplication" with the scalars from a division ring rather than a field is usually called a "module", not a "vector space".
Actually, Thomas Hungerford defines a vector space using division rings.

#### NonCommAlg

MHF Hall of Honor
Hi:
I have the following definition: let V be a vector space over
a division ring D. A mapping a of V into V is called a linear trans-
formation of V if it has the followiwng two properties:
(x+y)a = xa+ya for x,y $$\displaystyle \in$$ V,
(x$$\displaystyle \alpha$$)a = (xa)$$\displaystyle \alpha$$ for x $$\displaystyle \in$$ V, $$\displaystyle \alpha$$ D

And here I find an odd thing. If a is the mapping multiplication by
a scalar (that is, by an element of D), then a is not in general a linear trans-
formation of V according to the definition, because D needs not be
commutative. Any hint will be welcome.
multiplication by an element of D is a linear transformation iff that element is in the center of D.

• ENRIQUESTEFANINI

#### ENRIQUESTEFANINI

Actually, Thomas Hungerford defines a vector space using division rings.
And Neal H. McCoy too.

#### ENRIQUESTEFANINI

multiplication by an element of D is a linear transformation iff that element is in the center of D.
Quite understandable. And if D is a field the center of D is D. So in this case that mapping is always a linear transformation. Thanks a lot, NonCommAlg.