Let $F$ be a field and let $(K,*)$ be an associative $F$-algebra which, as a vector space, is finitely generated over $F$. Given an element $a\in K$, do there necessarily exist elements $a_1,a_2 \in K$ satisfying $a_1*a_2=a$?
I'm not sure how to go about this. Since we have an associative $F$-algebra we know that $v*(w*y)=(v*w)*y$ for all $v,w,y \in K$.
This depends on whether your definition of "associative $F$-algebra" includes the existence of a multiplicative identity. If it does, you can just take $a_1=1$ and $a_2=a$. If it doesn't, such an $a_1$ and $a_2$ need not exist. For instance, if $V$ is any vector space over $F$, you can make it a (non-unital) associative $F$-algebra by just defining $a_1*a_2=0$ for all $a_1,a_2\in V$, and then no such $a_1$ and $a_2$ exist unless $a=0$.