Exercise 1, § 2

Let A be a principal ideal domain, let M be a torsion A-module, let x, y be two elements of M, and let Aα and Aβ be the annihilators of x, y respectively; if δ is a gcd of α and β, show that the annihilator Ay of x + y is such that y divides the lcm αβ/δ of α and β, and is a multiple of $\alpha \beta / \delta^2$. When α and β are distinct powers $\pi^\mu, \pi^\nu$ of the same irreducible element π of A, show that $y = \pi^{\sup(\mu, \nu)}$.