Is there an infinite-dimensional, non-commutative complex local algebra $A$ (which is not a field) with the (unique) maximal left-ideal finitely generated as a left ideal? Or as a right ideal?



Local algebras with small maximal left ideals
Yes, the following should do the trick: 


Let $D$ be any skew field over $\mathbb{C}$. Then $D[[X]]$ is a local ring with maximal left ideal $(X)$ (see Lam, A first Course in noncommutative Rings, after 19.7) and $D[[X]]$ is clearly infinite-dimensional as $\mathbb C$-algebra. 


So it suffices to find $D$. According to Cohn's book quoted in 


$\qquad$[https://mathoverflow.net/questions/48173/infinite-dimensional-division-algebras](https://mathoverflow.net/questions/48173/infinite-dimensional-division-algebras-with-finite-center-and-their-involutions), 


$D$ can be choosen to be a twisted Laurent series ring over $\mathbb C$ (see Prop. 2.3.5 in [google books](http://books.google.de/books?id=u-4ADgUgpSMC&pg=PA70&lpg=PA70&dq=Skew+fields%3A+Theory+of+general+division+rings+2.3.5&source=bl&ots=TLmPbtRFfP&sig=6uyLBUgkxREExKu2dVHtr_TvaLg&hl=de&sa=X&ei=iN9fT8rEH4rYsgbr_fG7CQ&ved=0CDgQ6AEwAg#v=onepage&q&f=false)). 


