Let \(R\) be a commutative Noetherian ring. For an \(R\)-module \(M\), let \(\Omega_R(M)\) denotes the kernal of an onto homomorphism of \(R\)-modules \(P \rightarrow M\), with \(P\), a finitely generated projective \(R\)-module. \(\Omega_R(M)\) can be determined from \(M\), up to a projective summand. Define \(\Omega_R^d(M) := \Omega_R(\Omega_R^{d-1}(M))\) for any \(d \geq 1\). Over a local ring \((R,\mathfrak{m})\), an \(R\)-module \(M\) is said to be periodic if there exists \(n \in \mathbb{N}\) such that \(M \cong \Omega_R^n(M)\) and \(M\) is said to be eventually periodic if there exists an \(n \in \mathbb{N}\) and a non-negative integer \(\ell\), such that \(\Omega_R^{\ell}(M) \cong \Omega_R^{n+\ell}(M)\). In the paper under review, the author proves a structure theorem for \(J(R)\), the Grothendieck module of \(R\) (definition 2.1), when \(R\) is a Gorenstein local ring with the Krull-Remak-Schmidt property (Theorem 4.2). As a consequence, over a Gorenstein local ring \(R\) with the Krull-Remak-Schmidt property, that an \(R\)-module is eventually periodic if and only if its class in \(J(R)\) is annihilated by some non-zero element of \(\mathbb{Z}[t^{\pm 1}]\) and that \(R\) has a periodic module if and only if \(J(R)\) has non-zero torsion.