Morita Duality for the Rings of Generalized Power Series

Abstract

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. For any bimodule AMB, we show that the bimodule \( {}_{{\left[\kern-0.15em\left[ {A^{{S, \leqslant }} } \right]\kern-0.15em\right]}}{\left[ {M^{{S, \leqslant }} } \right]}_{{\left[\kern-0.15em\left[ {B^{{S, \leqslant }} } \right]\kern-0.15em\right]}} \) defines a Morita duality if and only if AMB defines a Morita duality and A is left noetherian, B is right noetherian. As a corollary, it is shown that the ring [[AS,≤]] of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule AMB such that B is right noetherian.