Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and let $X \boxtimes Y \subseteq \mathrm{End}_R(V\otimes W)$ be the set of those endomorphisms which are either of the form $x\otimes 1$ for some $x\in X$ or of the form $1\otimes y$ for some $y\in Y$.

Let $C_X \subseteq \mathrm{End}_R(V)$ be the algebra of endomorphisms which commute with all elements of $X$, and similarly define $C_Y$ and $C_{X \boxtimes Y}$

There is a canonical morphism of algebras $$C_X \otimes C_Y \xrightarrow{\qquad}C_{X \boxtimes Y}$$ sending $f\otimes g$ to the endomorphism $v\otimes w \mapsto f(v) \otimes g(w)$.

Under which condition on $R$ is this morphism always an isomorphism?

For example, if $R$ is a Dedekind domain, it is (in that case, $C_X$ is a direct factor of $\mathrm{End}_R(V)$ as an $R$--module).

Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and let $X \boxtimes Y \subseteq \mathrm{End}_R(V\otimes W)$ be the set of those endomorphisms which are either of the form $x\otimes 1$ for some $x\in X$ or of the form $1\otimes y$ for some $y\in Y$.

Let $C_X \subseteq \mathrm{End}_R(V)$ be the algebra of endomorphisms which commute with all elements of $X$, and similarly define $C_Y$ and $C_{X \boxtimes Y}$

There is a canonical morphism of algebras $$C_X \otimes C_Y \xrightarrow{\qquad}C_{X \boxtimes Y}$$ sending $f\otimes g$ to the endomorphism $v\otimes w \mapsto f(v) \otimes g(w)$.

Under which condition on $R$ is this morphism always an isomorphism?

For example, if $R$ is a Dedekind domain, it is (in that case, $C_X$ is a direct factor of $\mathrm{End}_R(V)$ as an $R$--module).