We consider monomial ideals of mixed product that are not normal in two sets of variables and we determine the powers that result complete. For all powers that are not complete we find a monomial tha lies in the integral closure of the power but it does not lie in the power.

Classification AMS: 13-XX, 05Cxx.

Let R be a commutative noetherian graded ring. We study SerreÂ’s condition and Ischebeck-AuslanderÂ’s condition in the graduate case. Let q be an integer, we characterize the * q-torsion freeness of graded modules on a * Gq-ring.