Fuzzy Maximal Sub-Modules

In this paper, we introduce and study the notions of fuzzy quotient module, fuzzy (simple, semisimple) module and fuzzy maximal submodule. Also, we give many basic properties about these notions.

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.

Semi-T-maximal sumodules

Let be a commutative ring with identity and be an -module. In this work, we present the concept of semi--maximal sumodule as a generalization of -maximal submodule. We present that a submodule of an -module is a semi--maximal (sortly --max) submodule if is a semisimple -module (where is a submodule of ). We investegate some properties of these kinds of modules.

L-Hollow modules

To consider R is a commutative ring with unity, be a nonzero unitary left R-module, is known hollow module if each proper submodule of is small. L-hollow module is a strong form of hollow module, where an R-module is known L-hollow module if has a unique maximal submodule which contains each small submodules. The current study deals with this class of modules and give several fundamental properties related with this concept. http://dx.doi.org/10.25130/tjps.24.2019.136

S-maximal Submodules

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ? W ? M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of rings and modules. .

ON δ-LOCAL MODULES AND AMPLY δ-SUPPLEMENTED MODULES

The first part of this paper investigates the structure of δ-local modules. We prove that the following statements are equivalent for a module M: (i) M is δ-local; (ii) M is a coatomic module with either (a) M is a semisimple module having a maximal submodule N such that N is projective and M/N is singular, or (b) M has a unique essential maximal submodule K ≤ M such that for every maximal submodule L ≠ K, M/L is projective. The second part establishes some properties of finitely generated amply δ-supplemented modules.

RINGS OVER WHICH FLAT COVERS OF SIMPLE MODULES ARE PROJECTIVE

Let R be a ring with identity. We prove that, the flat cover of any simple right R-module is projective if and only if R is semilocal and J(R) is cotorsion if and only if R is semilocal and any indecomposable flat right R-module with unique maximal submodule is projective.

MODULES WITH COINDEPENDENT MAXIMAL SUBMODULES

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if [Formula: see text] for every positive integer n and distinct maximal submodules L, Li (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.