It is easy to see that a ring with finitely many noncentral elements is either finite or commutative. The main topic of the paper under review is to study commutativity and near-commutativity in \(c^*\)-rings -- that is rings with the property that for each \(x\in R\), either \(x\) is periodic or there exists a positive integer \(K=K(x)\) such that \(x^k\in C\) for all \(k\geq K\). It is clear that every ring with finitely many noncentral elements is a \(c^*\)-ring and moreover, the class of \(c^*\)-rings is a much larger class, containing all periodic rings, all commutative rings, and all rings radical over a central ideal. The main results are the following: Theorem 1. Let \(R\) be a \(c^*\)-ring. Then (a) If \(R\) is reduced, then \(R\) is commutative. (b) If \(R\) is prime, then \(R\) is either commutative or periodic. Theorem 2. If \(R\) is a \(c^*\)-ring, then \(C(R)\), the commutator ideal of \(R\), is periodic. Moreover, if the set of nilpotent elements is commutative, then \(C(R)\) is nil. Theorem 3. If \(R\) is a torsion-free \(c^*\)-ring with 1, then \(R\) is commutative. Theorem 4. If \(R\) is a \(c^*\)-ring with 1 in which every nilpotent element is central, then \(R\) is commutative. Theorem 5. Let \(R\) be a ring in which each element is either periodic or central. If every nilpotent element is central, then \(R\) is commutative. This latter result whose proof is contained in the proof of Theorem 4 extends Herstein's well-known theorem that a periodic ring in which every nilpotent element is central, must be commutative.