Filtered Colimit of associative $k$-algebras that are domains

Question

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?

Answer 1

I don't know of any proof of this statement that would be easier for commutative algebras than for noncommutative algebras. If you have a diagram $F:C\to k\text{-Alg}$ where $C$ is filtered, $F(i)$ is a domain for each object $i$ of $C$, and $X$ is its colimit, suppose $x,y\in X$ are such that $xy=0$. Then there is some $i$ such that $x$ and $y$ are both lift to elements $\bar{x}$ and $\bar{y}$ of $F(i)$, and moreover such that $\bar{x}\bar{y}=0$. The fact that $F(i)$ is a domain now implies $\bar{x}=0$ or $\bar{y}=0$, so $x=0$ or $y=0$.