Let \(R\) be a Principal Ideal Domain (PID), and let \(r, s \in R \setminus \{0\}\) be non-zero elements. Determine the invariant factors of the \(R\)-module: \[M = R/(r) \oplus R/(s).\]
Solution
The module \[ M=R/(r)\oplus R/(s) \] has presentation matrix \[ A=\begin{pmatrix}r&0\\0&s\end{pmatrix}. \] Over a PID, the Smith normal form has invariant factors \(d_1\mid d_2\), determined up to associates by the determinantal divisors \[ \Delta_1=d_1, \qquad \Delta_2=d_1d_2. \] Here \[ \Delta_1\sim\gcd(r,s) \] because \(\Delta_1\) is a generator of the ideal generated by all entries of \(A\), and \[ \Delta_2\sim rs \] because the only \(2\times2\) minor is \(rs\). Therefore \[ d_1\sim\gcd(r,s), \qquad d_2\sim\frac{rs}{\gcd(r,s)}\sim\operatorname{lcm}(r,s). \] Thus, up to multiplication by units, the invariant factors are \[ \boxed{\gcd(r,s),\ \operatorname{lcm}(r,s)}. \] Equivalently, \[ M\cong R/(\gcd(r,s))\oplus R/(\operatorname{lcm}(r,s)). \]