Some results on relative dual Baer property

Abstract

Let R be a ring. In this article, we introduce and study relative dual Baer property. We characterize R-modules M which are RR-dual Baer, where R is a commutative principal ideal domain. It is shown that over a right noetherian right hereditary ring R, an R-module M is N-dual Baer for all R-modules N if and only if M is an injective R-module. It is also shown that for R-modules M1, M2,. . ., Mn such that Mi is Mj-projective for all i > j ∈ {1, 2,. .. , n}, an R-module N is n i=1 Mi-dual Baer if and only if N is Mi-dual Baer for all i ∈ {1, 2,. .. , n}. We prove that an R-module M is dual Baer if and only if S = EndR(M) is a Baer ring and IM = rM (lS(IM)) for every right ideal I of S.