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 $R_R$-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 $M_1$, $M_2$, $\ldots$, $M_n$ such that $M_i$ is $M_j$-projective for all $i > j \in \{1,2,\ldots, n\}$, an $R$-module $N$ is $\bigoplus_{i=1}^nM_i$-dual Baer if and only if $N$ is $M_i$-dual Baer for all $i\in \{1,2,\ldots,n\}$. We prove that an $R$-module $M$ is dual Baer if and only if $S=End_R(M)$ is a Baer ring and $IM=r_M(l_S(IM))$ for every right ideal $I$ of $S$.