Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains. 
Then the colimit of the obvious diagram is an integral domain.
Does this statement also hold in the case where we drop the commutativity condition?



Filtered Colimit of associative $k$-algebras that are domains
I don't see how commutativity matters.


Suppose $A$ is the filtered colimit of algebras $A\_i$ and $x,y\in A$ with $xy=0$. Then $x$ is represented by $x\_j\in A\_j$ and $y$ by $y\_k\in A\_k$ for some $j$ and $k$. By "filtered", these map to elements $x\_l,y\_l\in A\_l$ for some $l$ such that $x\_ly\_l=0$. Since $A\_l$ is a domain, either $x\_l=0$ (and therefore $x=0$) or $y\_l=0$ (and therefore $y=0$). 


