On free elementary \(\mathbb{Z}_p C_p\)-lattices (Q828615)

Let \(R = \mathbb{Z}_p C_p\) be the group ring of the cyclic group of order \(p\) over the localization of \(\mathbb{Z}\) at the prime \(p\). The authors shows that given two free \(R\)-modules \(M\) and \(L\) with \(pM \subseteq L \subseteq M\), there is an \(R\)-basis \((g_1,\dots,g_a)\) of \(M\) and \(0 \leq t \leq a\) such that \((g_1,g_2,\dots,g_t, pg_{t+1},\dots, pg_a)\) is an \(R\) basis of \(L\) showing that these lattices admit a compatible basis.