Serial piecewise domains

Abstract

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.