let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

The purpose of this paper is to introduce a new concept in module M over ring R, called e∗-essential submodule, which generalization an essential submodule. We will some examples and properties about such that, what the inverse image intersection submodules direct sum submodules. show relationship between submodule Noetherian R-module. Also we define e∗-closed with

The complement operation on modules has not been argued so far as we know. We observe that it plays an important role in software maintenance, as well as join and meet operations. A de nition of a module and operations on them is given in category theoretic terms; we adopt well-known characterization of complement as a left adjoint to join. The de nition is rst given locally and then extended t...

It has been observed by different authors that QTAG-modules behave very much like torsion abelian groups. In this paper, in section 3, we characterize quasi-essential submodules (Theorem 3.9) and further find a characterization for an h-pure submodule to be a direct summand (Theorem 3.11). In section 4, we obtained a necessary and sufficient condition for a submodule to be contained in a minima...

The cardinality of the minimal generating set of a module M i.e g(M) plays a very important role in the study of QTAG-Modules. Fuchs [1] mentioned the importance of upper and lower basic subgroups of primary groups. A need was felt to generalize these concepts for modules. An upper basic submodule B of a QTAG-Module M reveals much more information about the structure of M . We find that each ba...

primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. in fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  in this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

The notion of the square submodule of a module M over an arbitrary commutative ring R, which is denoted by RM, was introduced by Aghdam and Najafizadeh in [3]. In fact, RM is the R−submodule of M generated by the images of all bilinear maps on M. Furthermore, given a submodule N of an R−module M, we say that M is nil modulo N if μ(M×M) ≤ N for all bilinear maps μ on M. The main question about t...

Suppose that is an innnite set and k is a natural number. Let ] k denote the set of all k-subsets of and let F be a eld. In this paper we study the FSym(()-submodule structure of the permutation module F] k. Using the representation theory of nite symmetric groups, we show that every submodule of F] k can be written as an intersection of kernels of certain FSym(()-homomorphisms F] k ?! F] l for...

Submodule construction is the problem of finding a new submodule which, together with a given submodule, provides a behavior that conforms to a given desired global behavior. A new formulation of this problem and its solution in first-order logic is presented, and it is shown how the known solutions to this problem in the context of various communication paradigms and specification formalisms c...