Exercise 111.14.3. Let $R$ be a ring. Let $n \geq 1$. Let $A$, $B$ be $n \times n$ matrices with coefficients in $R$ such that $AB = f 1_{n \times n}$ for some nonzerodivisor $f$ in $R$. Set $S = R/(f)$. Show that

\[ \ldots \to S^{\oplus n} \xrightarrow {B} S^{\oplus n} \xrightarrow {A} S^{\oplus n} \xrightarrow {B} S^{\oplus n} \to \ldots \]

is exact.

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