Let A be a Noetherian unmixed local ring, \(I\subset A\) an ideal, \(S\subset A\) a multiplicative system such that \(I\cap S=\emptyset\) and \(I^{(n)}:=A\cap I^ nA_ S\), \(n\in {\mathbb{Z}}\). If \(\ell (I^{(n)})=ht(I^{(n)})\) for a certain \(n\geq 1\), \(\ell (I)\) denotes the analytic spread of I, then the ring \(R_ I:=\oplus_{n\geq 0}I^{(n)} \) is Noetherian. The converse is also true when \(A/I^{(n)}\) is Cohen- Macaulay for \(n>>0\). The first implication extends some results of \textit{D. Katz} and \textit{L. J. Ratliff} jun. [Commun. Algebra 14, 959-970 (1986; Zbl 0609.13011)], \textit{A. Ooishi} [Hiroshima Math. J. 15, 581-584 (1985; Zbl 0617.13011)] and others. Some nice examples show how important the ``unmixed'' condition is for the above result. Now, let A be a Noetherian normal local domain of dimension \(\geq 1\), \(I\subset A\) an ideal of height 1 and \(S:=A\setminus \cup p \), where the union is made over all \(p\in Spec(A)\), \(p\supset I\), \(ht(p)=1\). If the order t of the class of I in Cl(A) is finite then \(R_ I\) is Noetherian and the following assertions are equivalent: (1) \(R_ I\) is a CM-ring; (2) \(R_ I':=\oplus_{n\in {\mathbb{Z}}}I^{(n)} \) is a CM-ring; (3) \(G_ I:=\oplus_{n\geq 0}I^{(n)}/I^{(n+1)} \) is a CM-ring, (4) \(I^{(n)}\) is a maximal CM-module over A for \(0\leq n