  In the present paper, as a generalization of the classical periodic rings, we
explore those rings whose elements are additively generated by two (or more)
periodic elements by calling them additively periodic. We prove that, in some
major cases, additively periodic rings remain periodic too; this includes, for
instance, algebraic algebras, group rings, and matrix rings over commutative
rings. Moreover, we also obtain some independent results for the new class of
rings; for example, the triangular matrix rings retain that property.
