Some notes on modules in which all submodules have a unique closure

Modules Whose Small Submodules Have Krull Dimension

The main aim of this paper is to show that an AB5 module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension.

Some applications of the product of submodules in multiplication modules

Let R be a commutative ring with identity. Let N and K be two submodules of a multiplication R-module M. Then N=IM and K=JM for some ideals I and J of R. The product of N and K denoted by NK is defined by NK=IJM. In this paper we characterize some particular cases of multiplication modules by using the product of submodules.

Some Notes on Managing Closure Operators

It is widely known that closure operators on finite sets can be represented by sets of implications (also known as inclusion dependencies) as well as by formal contexts. In this paper we survey known results and present new findings concerning time and space requirements of diverse tasks for managing closure operators, given in contextual, implicational, or black-box representation. These tasks...

A note on primary-like submodules of multiplication modules

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...

Submodules of Secondary Modules