in this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.

We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings‎, ‎then $R/J(R)$ is nil-clean‎. ‎In particular‎, ‎under certain additional circumstances‎, ‎$R$ is also nil-clean‎. ‎These results somewhat improves on achievements due to Diesl in J‎. ‎Algebra (2013) and to Koc{s}an-Wang-Zhou in J‎. ‎Pure Appl‎. ‎Algebra (2016)‎. ‎...

‎The largest class of hyperstructures is the one which satisfies the weak properties; these are called $H_{v}$-structures‎. ‎In this paper we introduce a special product of elements in $H_{v}$-group $H$ and define a new class of $H_{v}$-groups called strongly $H_{v}$-groups‎. ‎Then we show that in strongly $H_{v}$-groups $beta=beta^{ast}$‎. ‎Also we express $theta$-hyperoperation and investigat...