Infinitely dimensional, affine nil algebras $A\otimes A^{op}$ and   $A\otimes A$ exist
  In this paper we answer two questions from [16], by showing that, over any algebraically closed field, $K$, there is a finitely generated, infinitely dimensional algebra $A$ such that algebras $A\otimes_{K}A$ and $A\otimes_{K} A^{op}$ are nil. 
Agata Smoktunowicz
math.RA