The Stacks project

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

is exact.