An elementary proof is given for Hutchinson’s duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

The goal of this article is to present the graded weakly S-primary ideals and g-weakly which are extensions primary ideals. We state P a ideal R if there exists s ∈ S such that for all x, y h(R), 0 6=xy P, then sx or sy Grad(P). Several properties characteristics as well investigated.

in this paper we characterize the radical of an arbitrary‎ ‎submodule $n$ of a finitely generated free module $f$ over a‎ ‎commutatitve ring $r$ with identity‎. ‎also we study submodules of‎ ‎$f$ which satisfy the radical formula‎. ‎finally we derive‎ ‎necessary and sufficient conditions for $r$ to be a‎ ‎pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $rbigopl...

The aim of this paper is to introduce and study two new classes of functions called weakly $e^{*}$-open functions and weakly $e^{*}$-closed functions via the concept of $e^{*}$-open set defined by Ekici cite{erd1}. The notions of weakly $e^{*}$-open and weakly $e^{*}$-closed functions are weaker than the notions of weakly $beta$-open and weakly $beta$-closed functions defined by Caldas and Nava...

After the paper [1] was published, following information about D-extending modules pointed out by Adnan Tercan. Let $r$ be a left exact preradical in category of right $R$-modules and let $M$ $R$-module. A submodule $N$ is called provided $r(M/N)=0$. module an ES-module every direct summand $M$. ES-modules introduced A. Tercan [2]. Since socle preradical, coincide with for $r=Soc$.The author wh...