For any prime \(p\) and integer \(r\ge1\), \(\mathbb{Z}_{p^r}\) is a finite ring and any element \(a\in \mathbb{Z}_{p^r}\) is of the form \(a=\alpha_0+\alpha_1p+\cdots+\alpha^{r-1}p^{r-1}\), \(\alpha_i\in\{0,1,2,\ldots, p-1\}\) for \(0\le i\le r-1\). A convolution code \(\mathcal{C}\) over the ring \(\mathbb{Z}_{p^r}\) can be defined as a free \(\mathbb{Z}_{p^r}\)-submodule of \(\mathbb{Z}_{p^r}^n\). In this article, authors study convolution codes described by the linear system \(\Sigma=(A,B,C,D)\) given by the input-state-output representations \((x_{t+1}=x_tA+u_tB; y_t=x_tC+u_t D)\), where \(x_t\)-the state vector with \(x_0=0\), \(u_t\)-the input, \(y_t\)-the output for at any time \(t\) and \(A,B,C,D\) are matrices over \(\mathbb{Z}_{p^r}\). They show that the set of finite weight input-state-output trajectories of \(\Sigma\) that are polynomials over \(\mathbb{Z}_{p^r}\) has the structure of a free \(\mathbb{Z}_{p^r}\)-submodule of \(\mathbb{Z}_{p^r}^n\), a convolution code \(\mathcal{C}(A,B,C,D)\) with finite support.