Let $R_1,R_2$ be two (left) artinian rings (not necessarily commutative), is $R_1\times R_2$ necessarily artinian ?
I also have another related question that came to my head while thinking about the first one. If $R_1,R_2$ both have a finite number of left ideals, must $R_1\times R_2$ have a finite number of left ideals too ?
Thank you
The answer to both questions is yes because the left ideals of $R_1\times R_2$ have the form $I_1\times I_2$ where $I_i$ is a left ideal of $R_i$.
This is not difficult to prove: for $I\lhd R$, just consider the left ideals $R(1,0)I$ and $R(0,1)I$.