  Let $\Lambda$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa
algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of
totally real fields. When the non-primitive Iwasawa module over the cyclotomic
$\mathbb{Z}_p$-extension has a free resolution of length one over the group
ring $\Lambda[G]$, we prove that the validity of the non-commutative Iwasawa
main conjecture allows us to find a representative for the non-primitive
$p$-adic $L$-function (which is an element of a $K_1$-group) in a maximal
$\Lambda$-order. This integrality result involves a careful study of the
Dieudonn\'e determinant. Using a cohomolgoical criterion of Greenberg, we also
deduce the precise conditions under which the non-primitive Iwasawa module has
a free resolution of length one. As one application of the last result, we
consider an elliptic curve over $\mathbb{Q}$ with a cyclic isogeny of degree
$p^2$. We relate the characteristic ideal in the ring $\Lambda$ of the
Pontryagin dual of its non-primitive Selmer group to two characteristic ideals,
viewed as elements of group rings over $\Lambda$, associated to two
non-primitive classical Iwasawa~modules.
