Tensor product of certain algebras

Question

If S is an $R$-algebra of finite dimension, and $A$ is an algebra of infinite dimension, then is $S \otimes S \otimes A \cong S \otimes A$?

Answer 1

No. For example, let $k$ be a ground field, $S=M_2(k)$ and $A=k(t)$, the field of rational functions in the variable $t$. Then $S\otimes S\otimes A\cong M_4(k(t))$ and $S\otimes A\cong M_2(k(t))$. There two algebras are not isomorphic (not even as rings)

For example, each of them is a direct sum of minimal left ideals, and the numbers of summands are different. Or: their dimensions over their centers are different.

Another example: let $\mathbb Q$ be our base field, $S=\mathbb Q(\sqrt 2)$ and $A=\mathbb R$.