Quasi regular modules and trivial extension

Recall that a ring $R\ $is said to be a quasi regular ring if its total quotient ring $qinsert ignore into journalissuearticles values(R);\ $is \textit{von Neumann regular}. It is well known that a ring $R\ $is quasi regular if and only if it is a reduced ring satisfying the property: for each $a\in R,$ $ann_{R}insert ignore into journalissuearticles values(ann_{R}insert ignore into journalissuearticles values(a););=ann_{R}insert ignore into journalissuearticles values(b);$ for some $b\in R$. Here, in this study, we extend the notion of quasi regular rings and rings which satisfy the aforementioned property to modules. We give many characterizations and properties of these two classes of modules. Moreover, we investigate the insert ignore into journalissuearticles values(weak); quasi regular property of trivial extension.