Division Rings and Division Algebras On wikipedia (division algebra page), it says "Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field."
I've just been through a proof this theorem, but this proof proves that if D is a finite division ring then it is a finite field. Is wikipedia wrong, or does this amount to the same thing?
What I mean is, does the fact that any finite division ring is a field imply that any finite division algebra is a field?
If the above were true, it would imply that for any division ring D, if D forms an algebra then this algebra is a division algebra.
I feel like my trouble understanding this stems from a lack of understanding of vector spaces. If I have a set S and it forms a vector space over a field F, can F-linear combinations of elements of S be outside of S? If they can't, then I think my problem is solved.
Apologies for the poorly phrased question.
 A: There's actually no difference between (finite) division rings and (finite) division algebras.
If $D$ is a finite division ring, then its prime subring $P$ is a (finite) field, because $D$ has no zero divisors. Thus $D$ can be considered as a $P$-algebra. Of course $P\cong\mathbb{Z}/p\mathbb{Z}$ where $p$ is the characteristic of $D$.
Similarly, if $D$ is an infinite division ring, its prime subring $P$ is either a finite field (if $D$ has nonzero characteristic) or $\mathbb{Z}$. In this case, since $D$ is a division ring, the inverses of the nonzero elements of $P$ belong to $D$, so they form, together with $P$ a field isomorphic to $\mathbb{Q}$. It's easy to see that if a nonzero element of $D$ is in the center, then its inverse is in the center too.
A: Theorem of Wedderburn: Every finite division ring (in particular, every finite division algebra) is a field.
You can find a proof in:
I. N. Herstein, Noncommutative Rings, 1971, p.69,
Lam T.Y. A first course in noncommutative ring theory, 1991, p.214,
Richard S. Pierce, Associative algebras. Springer-Verlag, 1982.
In the last book associative algebras over a commutative ring are considered, so rings are included in consideration.
