Topologically constrained molecular systems are relevant to many areas of physics and biology, from the collapse of a polymer gel to the existence of chromosome territories. As a model system for the study of such unusual interactions, we simulated melts of unconcatenated polymer rings, where topology directly influences not only dynamic properties, as is the case for linear chains, but also the statics. In order to access the relatively large chain lengths required to observe significant effects, we implemented an efficient, on-lattice Monte Carlo model allowing for fast relaxation thanks to non-local moves which randomly displace kinks along the polymer contour. The sufficiently long rings are shown to behave as compact objects, their size scaling as N(1/3) for large chain length N; this observation is supported by measurements of several other static properties. The entanglement length of the rings' linear counterparts is used to characterize the topological constraints, allowing for an estimate of the onset of the regime where rings are compact.