The topic of the talk is motivated by certain questions on analytic side of (non-commutative) Iwasawa theory, namely the integrality properties of the p-adic L-function. Though we are motivated by these properties on the analytic side of Iwasawa theory, our work deals with the algebraic objects, called Iwasawa modules. We will discuss certain (non-primitive) Iwasawa modules that have a free resolution of length one over appropriate Iwasawa algebras. This is joint work with Alexandra Nichifor.