Siberian Federal University

Let A = {Ar | r ∈ Φ} be a carpet of additive subgroups of type Φ over an arbitrary commu- tative ring K. A sufficient condition for the carpet A to be closed is established. As a corollary, we obtain a positive answer to question 19.63 from the Kourovka notebook and a confirmation of one conjecture by V. M. Levchuk, provided that the type of Φ is different from Cl, l > 5 when the characteristic of the ring K is 0 or 2m for some natural number m > 1. Also, a partial answer to question 19.62 has been obtained