Let $R$ be a commutative ring and $M$ be an $R$-module. A submodule $N$ of $M$ is called a d-submodule $($resp., an fd-submodule$)$ if $\ann_R(m)\subseteq \ann_R(m')$ $($resp., $\ann_R(F)\subseteq \ann_R(m'))$ for some $m\in N$ $($resp., finite subset $F\subseteq N)$ and $m'\in M$ implies that $m'\in N.$ Many examples, characterizations, and properties of these submodules are given. Moreover, we use them to characterize modules satisfying Property T, reduced modules, and von Neumann regular modules.
Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of [Formula: see text]-ideals. A proper ideal [Formula: see text] of [Formula: see text] is said to be a [Formula: see text]-ideal if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some basic properties of [Formula: see text]-ideals are studied. For instance, we give a method to construct [Formula: see text]-ideals that are not [Formula: see text]-ideals. Among other things, it is shown that if [Formula: see text] admits a [Formula: see text]-ideal that is not an [Formula: see text]-ideal, then [Formula: see text] is a local ring. Also, we provide a necessary and sufficient condition in term of [Formula: see text]-ideals for a ring to be a total quotient ring and we determinate the [Formula: see text]-ideals of a chained ring. Finally, we give an idea about some [Formula: see text]-ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of [Formula: see text]-ideals.
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