Let $R$ be an artinian algebra and $S$ be a finite dimensional algebra over the field $k$. How can i show that $R\otimes_kS$ is artinian?
I know that $S$ is also artinian since it is finite dimensional. If the free $k-$module $F$ with basis $R\times S$ is artinian then so does the tensor product because it is a quotient of $F$. But i guess it is not true. On the other hand i could show that $R\otimes_kS$ has DCC. In this case i need what do ideals of the tensor pruduct look like. Is there any certain description of them?
Thank you...
Hints:
Lemma: Let $k$ be a field and $A$ and $B$ be $k$ algebras. Then $A\otimes_k B$ is a left $A$ module under the action $a\cdot x:= (a\otimes 1)x$.
Lemma: Let $R$ be any commutative ring, and $A$ be an $R$ algebra. Then the left $A$ module $A\otimes_R R^n\cong A^n$.
Remark: A descending chain of ideals in $R\otimes_k S$ would also be a descending chain of left $R$ modules when you view $R\otimes_k S$ as a left $R$ module...