Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

‎In this paper we give an upper bound for Noetherian dimension of all injective modules with Krull dimension on arbitrary rings‎. ‎In particular‎, ‎we also give an upper bound for Noetherian dimension of all Artinian modules on Noetherian duo rings.

we introduce the notions of t-dual rickart and strongly t-dual rickart modules. we provide several characterizations and investigate properties of each of these concepts. it is shown that every free (resp. finitely generated free) $r$-module is t-dual rickart if and only if $overline{z}^2(r)$ is a  direct summand of $r$ and end$(overline{z}^2(r))$ is a semisimple (resp. regular) ring. it is sho...

an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dim...

An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

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