Matrix and Polynomial Reversibility of Rings

Reversibility of rings is a generalization of commutativity, but more than often, this weaker commutativity is a consequence of the absence of certain zero products. For example, a reversible ring is prime if and only if it is an integral domain, and a ring is reduced if and only if it is reversible and semiprime. Here we define and investigate classes of more restricted reversible rings which fulfill stronger commutative requirements, for example, rings that satisfy ab = 0 = ac + db implies ba = 0 = ca + bd.