How to show that $ \left(k[\textbf{x}]/I\right)\otimes_k\left(k[\textbf{y}]/J\right)\cong k[\textbf{x},\textbf{y}]/(I^e+J^e)$?
Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?