Krull modules and completely integrally closed modules

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

On Fully Closed Stable Modules

In this paper, we studied the notion of the fully closed stable module and identified some basic properties of this notion. We also investigated some concepts which are related to this module. In addition, the notion of CL-duo and fully closed stable modules were also studied. HIGHLIGHTS Studying the concept of fully closed stable modules Connect two concepts with important algebraic properties Giving new results to related concepts such as duo module closed multiplication module and closed monomorphism coretractable module

Arithmetic modules over generalized Dedekind domains

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

On the compressed essential graph of a module over a commutative ring

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module. The compressed essential graph of [Formula: see text], denoted by [Formula: see text] is a simple undirected graph associated to [Formula: see text] whose vertices are classes of torsion elements of [Formula: see text] and two distinct classes [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, we study diameter and girth of [Formula: see text] and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that [Formula: see text], whenever [Formula: see text] is Noetherian and [Formula: see text] is a finitely generated multiplication module with [Formula: see text].

Relationship of Essentially Small Quasi-Dedekind Modules with Scalar and Multiplication Modules

Let be a ring with 1 and D is a left module over . In this paper, we study the relationship between essentially small quasi-Dedekind modules with scalar and multiplication modules. We show that if D is a scalar small quasi-prime -module, thus D is an essentially small quasi-Dedekind -module. We also show that if D is a faithful multiplication -module, then D is an essentially small prime -module iff is an essentially small quasi-Dedekind ring.

Implementation of Montgomery Multiplier using Scalable Architecture

This paper describes the methodology and design of a scalable Montgomery multiplication module. This multiplier can manipulate any number of bits without any limitation. The size of a word depends upon the area which is available and also the performance which is required. After the general architecture is described, hardware organization is analyzed for implementing parallel computation and the discussions on design tradeoffs are done for recognising best configuration for hardware.

On Beta-Prime Submodules

We introduce the concepts of $\beta$-prime submodules and weakly $\beta$-prime submodules of unital left modules over a commutative ring with nonzero identity. Some properties of these concepts are investigated. We use the notion of the product of two submodules to characterize $\beta$-prime submodules of a multiplication module. Characterization of $\beta$-prime and weakly $\beta$-prime submodules of arbitary modules are also given.

Choice of Confounding in the 2k Factorial Design in 2b Blocks

In 2k complete factorial experiment, the experiment must be carried out in a completely randomized design. When the numbers of factors increase, the number of treatment combinations increase and it is not possible to accommodate all these treatment combinations in one homogeneous block. In this case, confounding in more than one incomplete block becomes necessary. In this paper, we considered the choice of confounding when k > 2. Our findings show that the choice of confounding depends on the number of factors, the number of blocks and their sizes. When two more interactions are to be confounded, their product module 2 should be considered and thereafter, a linear combination equation should be used in allocating the treatment effects in the principal block. Other contents in other blocks are generated by multiplication module 2 of the effects not in the principal block. Partial confounding is recommended for the interactions that cannot be confounded.

On a generalization of prime submodules of a module over a commutative ring

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.