On the primefulness of local cohomology modules

Let be a commutative Noetherian ring with identity For a non-zero module . We prove that a multiplication primeful module and are I-cofinite and primeful, for each where is an ideal of with . As a consequence, we deduce that, if and are multiplication primeful R- modules, then is primeful. Another result is, for a projective module over an integral domain, admits projective resolution such that each is primeful (faithfully flat).

The New Results in n-injective Modules and n-projective Modules

In this paper, we introduce and clarify a new presentation between the n-exact sequence and the n-injective module and n-projective module. Also, we obtain some new results about them.

Copure direct injective modules

In this paper, we have introduced copure-direct-injective modules. A right [Formula: see text]-module [Formula: see text] is said to be copure-direct-injective if every copure submodule of [Formula: see text] isomorphic to a direct summand of [Formula: see text] is itself a direct summand. We have studied properties of copure-direct-injective modules. We characterized rings over which every (cofinitely generated, free, projective) module is copure-direct-injective. We have examined for which rings or under what conditions copure-direct-injective modules are direct-injective, quasi-injective, copure-injective, injective. Also we have compared copure-direct-injective modules with pure-direct-injective modules.

Strongly D2 modules and their applications

An [Formula: see text]-module [Formula: see text] is called strongly [Formula: see text] if [Formula: see text] is a [Formula: see text] (equivalently, direct projective) module for every positive integer [Formula: see text]. In this paper, we consider the class of quasi-projective [Formula: see text]-modules, the class of strongly [Formula: see text] [Formula: see text]-modules and the class of [Formula: see text]-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly [Formula: see text] modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly [Formula: see text] modules.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

Schanuel's Lemma in P-Poor Modules

Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. In module theory there is a concept about projective module, i.e. a module over ring R in which it is projective module relative to all modules over ring R. Next, there is the fact that every module over ring R is projective module relative to all semisimple modules over ring R. If P is a module over ring R which it’s projective relative only to all semisimple modules over ring R, then P is called p-poor module. In the discussion of the projective module, there is a lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that is isomorphic to . That lemma is known as Schanuel’s lemma in projective modules. Because the p-poor module is a special case of the projective module, then in this paper will be discussed about Schanuel’s lemma in p-poor modules

Characterisation and applications of $\Bbbk$-split bimodules

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk $-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk $-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk $-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk [x,y]/(x^2,y^2,xy)$.