From the paper: Adapting Azumaya’s exactness to bimodules, Camillo, Fuller, and Haack said that a bimodule that has a composition series whose composition factors are balanced is an exact bimodule and that a ring R is an exact ring in case the regular bimodule \({}_ RR_ R\) is exact. We call a one-sided module \({}_ RM\) exact if the bimodule \({}_ RM_{End(_ RM)}\) is an exact bimodule. After observing some properties of exact modules, we show that the exactness for rings and reflexive modules is preserved under Morita duality, and that a basic exact artinian ring and the endomorphism ring of its minimal cogenerator have the same Loewy series. This lends support to Azumaya’s conjecture that exact artinian rings have self-duality. Our concluding theorem proves that every module over a serial ring is exact.