The annihilators comaximal graph

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

On Small Primary Modules

Let be a commutative ring with an identity and be a unitary -module. We say that a non-zero submodule of is primary if for each with en either or and an -module is a small primary if = for each proper submodule small in. We provided and demonstrated some of the characterizations and features of these types of submodules (modules).

Sifat-sifat Submodul Prima dan Submodul Prima Lemah

Penelitian ini merupakan penelitian kajian pustaka yang bertujuan untuk mengkaji sifat-sifat submodul prima dan submodul prima lemah serta hubungan antara keduanya. Kajian dimulai dari definisi submodul prima dan submodul prima lemah, selanjutnya dikaji mengenai sifat-sifat dari keduanya. Pada penelitian ini, semua ring yang diberikan adalah ring komutatif dengan unsur kesatuan dan modul yang diberikan adalah modul uniter. Sebagai hasil dari penelitian ini diperoleh beberapa pernyataan yang ekuivalen, misalkan suatu -modul , submodul sejati di dan ideal di , maka ketiga pernyataan berikut ekuivalen, (1) merupakan submodul prima, (2) Setiap submodul tak nol dari -modul memiliki annihilator yang sama, (3) Untuk setiap submodul di , subring di , jika berlaku maka atau . Di lain hal, pada submodul prima lemah jika diberikan suatu -modul, submodul sejati di , maka pernyataan berikut ekuivalen, yaitu (1) Submodul merupakan submodul prima lemah, (2) Untuk setiap , jika maka . Selain itu, didapatkan pula hubungan antara keduanya, yaitu setiap submodul prima merupakan submodul prima lemah.Kata Kunci: Submodul Prima, Submodul Prima Lemah, Ideal Prima. This research is literature study that aims to examine the properties of prime submodules and weakly prime submodules and the relationship between both of them. The study starts from the definition of prime submodules and weakly prime submodules, then reviewed about the properties both of them. Throughout this paper all rings are commutative with identity and all modules are unitary. As the result of this research, obtained several equivalent statements, let be a -module, be a proper submodule of and ideal of , then the following three statetments are equivalent, (1) is a prime submodule, (2) Every nonzero submodule of -module has the same annihilator, (3) For any submodule of , subring of , if then or . In other case, for weakly prime submodules, if given is a unitary -module, be a proper submodule of , then the following statements are equivalent, (1) is a weakly prime submodule, (2) For any , if then . In addition, also found the relationship between both of them, i.e. any prime submodule is weakly prime submodule.Keywords: Prime Submodules, Weakly Prime Submdules, Prime Ideal.

The Intersection Property of Annihilators of Both Modules and Maximal Submodules Belonging to the Same Module

During that article T stands for a commutative ring with identity and that S stands for a unitary module over T. The intersection property of annihilatoers of a module X on a ring T and a maximal submodule W of M has been reviewed under this article where he provide several examples that explain that the property. Add to this a number of equivalent statements about the intersection property have been demonstrated as well as the direct sum of module that realize that the characteristic has studied here we proved that the modules that achieve the intersection property are closed under the direct sum with a specific condition. In addition to all this, the relationship between the modules that achieve the above characteristics with other types of modules has been given.

I-Nearly Prime Submodules

Let be a commutative ring with identity, and a fixed ideal of and be an unitary -module. In this paper we introduce and study the concept of -nearly prime submodules as genrealizations of nearly prime and we investigate some properties of this class of submodules. Also, some characterizations of -nearly prime submodules will be given.

Approximaitly Quasi-Prime Submodules and Some Related Concepts

“Let be a commutative ring with identity and is a left unitary -module. A proper submodule of is called a quasi-prime submodule, if whenever , where , implies that either or ”. As a generalization of a quasi-prime submodules, in this paper we introduce the concept of approximaitly quasi-prime submodules, where a proper submodule of is an approximaitly quasi-prime submodule, if whenever , where , implies that either or , where is the intersection of all essential submodules of . Many basic properties, characterization and examples of this concept are given. Furthermore, we study the behavior of approximaitly quasi-prime submodules under -homomorphisms. Finally, we introduced characterizations of approximaitly quasi-prime submodule in class of multiplication modules.

On FGDF-modules

Let R be a unital ring and M a unitary module not necessary over R. The FGDF-module is a generalization of FGDF-rings (Touré, Diop, Mohamed and Sangharé, 2014). In this work, we first give some properties of FGDF-modules. After that, we show that for a finitely generated module M, M is a FGDF-module if and only if M is of finite representation type module. Finally, we show that M is a finitely generated FGDF-module if and only if every Dedekind finite module of $\sigma[M]$ is noetherian.

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Let R be a commutative ring with identity, and let M be a unitary module over R. We call M H-smaller (HS for short) if and only if M is infinite and |M/N| < |M| for every nonzero submodule N of M. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose M is faithful over R, R is a domain (we will show that we can restrict to this case without loss of generality), and K is the quotient field of R. If M is HS over R, then R is HS as a module over itself, R ⊆ M ⊆ K, and there exists a generating set S for M over R with |S| < |R|. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

A note on universally zero-divisor rings

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.