On some commutativity-or-periodicity conditions for rings

We study rings R with the following properties: (a) every proper noncentral subring is periodic; (b) every proper noncentral subring of zero divisors is periodic. We show that rings with property (a) must be either commutative or periodic, and that rings with property (b) which satisfy certain additional hypotheses are either periodic or have all zero divisors central. We also pose a question on the existence of a special kind of ring with property (b).