Let $k$ be a commutative ring and $R$ be a commutative $k$-algebra. Denote $\otimes=\otimes_k$. Let $A,B,C,D$ be $R$-modules.
Is it true that $(A\otimes B) \otimes_R (C \otimes D) \cong (A\otimes_R C) \otimes (B\otimes_R D)$? I have a map from the left to the right but I'm not sure it's an iso.
No. Take $ k = \mathbb R $, $ R = \mathbb C $, and $ A = B = C = D = \mathbb C $. Then,
$$ (A \otimes B) \otimes_R (C \otimes D) \cong \mathbb C^2 \otimes_\mathbb C \mathbb C^2 \cong \mathbb C^4 $$
whereas
$$ (A \otimes_R C) \otimes (B \otimes_R D) \cong \mathbb C \otimes_\mathbb R \mathbb C \cong \mathbb C^2 $$