It is well-know that flat base change in general does not preserve injective modules, i.e. let $M$ be an injective $R$-module and $R\to S$ be a flat morphism, then it is not necessary that $M\otimes_R S$ is an injective $S$-module.
Now let's consider a more special case:
Let $R$ be a commutative noetherian algebra over a base field $k$ and $l/k$ be a (non-finite) algebraic field extension. Let $M$ be an injective $R$-module. Then is it necessary that $M\otimes_kl$ is an injective $R\otimes_kl$-module?
Notice that the answer is negative if $l/k$ is not algebraic as pointed out by @EricWofsey in the answer of this stack exchange question.