In the course notes about linear algebra, my professor gives a proof of matrix multiplication on some ring $A$:
$$ \det(BC) = \det(B) \det(C) $$ where $B,C \in M_{n}(A)$ given that the result has been known for any field $K$ (i.e. where $B,C \in M_{n}(K))$.
The proof as follows: we consider $B,C$ as formal matrices: $B = (B_{ij})_{1 \leq i,j \leq n}$ and $C = (C_{kl})_{1 \leq i,j,k,l \leq n}$ (so $B_{ij}$ and $C_{kl}$ are indeterminates), then consider the cannonical injection: $$ \mathbb{Z} [B_{ij},C_{kl} ] \hookrightarrow \mathbb{Q}(B_{ij},C_{kl}) $$ Since $\mathbb{Q}(B_{ij},C_{kl})$ is a field, the matrix multiplication is already confirmed: $$ \det (BC) = \det(B) \det(C) $$ But because $B,C \in \mathbb{Z} [ B_{ij},C_{kl}]$ then this equality happens in the ring $\mathbb{Z} [ B_{ij},C_{kl}]$. To obtain the result, consider a specialization: $$ \phi: M_n({\mathbb{Z}[B_{ij},C_{kl}]}) \to M_{n}(A) $$ which maps $B_{ij} \to m_{ij} \in A$ for some matrix $B = (m_{ij})_{1 \leq i,j \leq n} \in M_{n}(A)$ and $C_{kl} \to m_{kl} \in A$ for some matrix $C = (m_{kl})_{1 \leq k,l \leq n} \in M_{n}(A)$, then $\phi$ is a ring homomorphism, and we have the result.
This technique is used in several proofs of the course notes (once the prof needs to prove some results on ring which have been known for fields).
For me, the proof seems very artificial. Do you know any sources (books,etc.) to learn more about this?
Thanks for any hint.