Let $R$ be a commutative ring with identity and $M$ be an $R$-module. It is shown that the usual lattice $\mathcal{V}(_{R}M)$ of varieties of submodules of $M$ is a distributive lattice. If $M$ is a semisimple $R$-module and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are lattice homomorphisms. It is shown that if $M$ is a faithful primeful $R$-module, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic lattices, and therefore $\mathcal{V}(_{R}M)$ and the lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras.

Inspired in [7],  ‎we strengthened semiperfect modules to $\mathcal{LA}$-semiperfect modules by adding the concept of locally artinian submodule and we proved that the concept of these modules is not empty‎. ‎Then we characterized these projective modules with the help of defining $\mathcal{LA}$-projective covers‎. ‎Thus we achived the necessary and sufficient condition for the projective module between the notion of $\mathcal{LA}$-semiperfect modules and the notion of locally artinian supplemented modules.

‎The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics‎, ‎where group rings play the main role‎. ‎The group ring $\rga$ is actually an associative ring that inherits the properties of the group $\ga$ and the ring of coefficients $R$‎. ‎In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory‎, ‎it also has applications in algebraic topology‎, ‎homological algebra‎, ‎algebraic K-theory and algebraic coding theory‎.‎In this article‎, ‎we provide a complete description of Gorenstein flat-cotorsion modules over the group ring $\rga$‎,‎where $\ga$ is a group and $R$ is a commutative ring‎. ‎It will be shown that if $\ga'\leqslant \ga$ is a finite-index subgroup‎, ‎then the restriction of scalars along the ring homomorphism $\rga'\rt\rga$ as well as its right adjoint $\rga\otimes_{\rga'}-$‎, ‎preserve the class of Gorenstein flat-cotorsion modules‎. ‎Then‎, ‎as a result‎, ‎Serre's Theorem is proved for the invariant $\Ghcd_{R}\ga$‎, ‎which refines the Gorenstein homological dimension of $\ga$ over $R$‎, ‎$\Ghd_{R}\ga$‎, ‎and is defined using flat-cotorsion modules‎. ‎Moreover‎, ‎we show that the inequality $\GF (\rga)\leqslant \GF (R)+{\cd_{R}\ga}$ holds for the group ring $\rga$‎, ‎where $\GF (R)$ denotes the supremum of Gorenstein flat-cotorsion dimensions of all $R$-modules and $\cd_{R}\ga$ is the cohomological dimension of $\ga$‎ over $R$‎.

‎The purpose of this paper is to define and investigate the notion of quasi $z^\circ$-submodules of modules over a  commutative ring as an extension of $z^\circ$-ideals of commutative rings. Also, we obtain some related results when $M$ is a reduced multiplication $R$-module.

‎In this paper‎, ‎our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$‎. ‎Let $T$ be an arbitrary submodule of an $R$-module $M$‎, ‎then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎whenever for every submodule $X$ of $M$‎, ‎$N\cap X\subseteq T$ implies that‎ ‎$(T:M)\subseteq ^{e}{\rm Ann}(X)$‎. ‎We investigate some new results concerning to this class of submodules‎. ‎Among various results we prove that for a faithful multiplication $R$-module $M$‎, ‎if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$‎. ‎The converse is true if $M$ is moreover a finitely generated module‎.

‎A sequence of continuous linear mappings $\{\Phi_n\}_{n=0}^\infty$ form a Hilbert C* -‎module‎ M into M is called a local higher derivation if for each $a\in\mathfrak{M}$ there is a continuous higher derivation $\{\varphi_{a,n}\}_{n=0}^\infty$ ‎on‎ M such that $\Phi_n(a)=\varphi_{a,n}(a)$ for each non-negative integer n‎. ‎In this paper w‎e show that if M is a Hilbert C* -‎module‎ such that every local derivation on M is a derivation, then each local higher derivation on M‎ is a higher derivation‎. Also, we prove that each local higher derivation on a unital C*-algebra is automatically continuous‎.