TITLE: What is a split $\mathbb{K}$-algebra?
QUESTION [6 upvotes]: After some considerations the article I'm reading concludes: "...hence H is a simple split $\mathbb{K}$-algebra".
I can't find this definition anywhere: what does "split" mean?

REPLY [4 votes]: According to this note, a split $k$-algebra $A$ is a $k$-algebra all of whose simple modules $M$ are absolutely simple, in the sense that after extension of scalars $M \mapsto M \otimes_k L$ (where $k \to L$ is a field extension) they remain simple. 
If $A$ is semisimple, I believe this condition is equivalent to the condition that $A$ is a finite product of matrix algebras $M_n(k)$. And if $A$ is (finite-dimensional and) simple, I believe this condition is equivalent to the condition that $A$ is a matrix algebra $M_n(k)$. 
The terminology is, as far as I know, motivated by a generalization of the notion of splitting field of a polynomial. A splitting field for a $k$-algebra is a field extension $k \to L$ such that $A \otimes_k L$ is split, and in the special case $A = k[x]/f(x)$ this is the same thing as a splitting field of $f(x)$.