Serial piecewise domains

A ring $A$ is called a piecewise domain with respect to the complete set of idempotents $\{e_1, e_2,\ldots, e_m\}$ if every nonzero homomorphism $e_iA \rightarrow e_jA$ is a monomorphism. In this paper we study the rings for which conditions of being piecewise domain and being hereditary (or semihereditary) rings are equivalent. We prove that a serial right Noetherian ring is a piecewise domain if and only if it is right hereditary. And we prove that a serial ring with right Noetherian diagonal is a piecewise domain if and only if it is semihereditary.