Modules whose injectivity domains are restricted to semi-artinian modules

We introduce the notion of semi-poor modules and consider the possibility that all modules are either injective or semi-poor. This notion gives a generalization of poor modules that have minimal injectivity domain. A module [Formula: see text] is called semi-poor if whenever it is [Formula: see text]-injective and [Formula: see text], then the module [Formula: see text] has nonzero socle. In this paper the properties of semi-poor modules are investigated and are used to characterize various families of rings. We introduce the rings over which every module is either semi-poor or injective and call such condition property [Formula: see text]. The structure of the rings that have the property [Formula: see text] is completely determined. Also, we give some characterizations of rings with the property [Formula: see text] in the language of the lattice of hereditary pretorsion classes over a given ring. It is proved that a ring [Formula: see text] has the property [Formula: see text] iff either [Formula: see text] is right semi-Artinian or [Formula: see text] where [Formula: see text] is a semisimple Artinian ring and [Formula: see text] is right strongly prime and a right [Formula: see text]-ring with zero right socle.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

A Cohen-type theorem for w-Artinian modules

Let [Formula: see text] be a commutative ring with identity. In this paper, a Cohen-type theorem for [Formula: see text]-Artinian modules is given, i.e. a [Formula: see text]-cofinitely generated [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if [Formula: see text] is [Formula: see text]-cofinitely generated for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. As a by-product of the proof, we also obtain a detailed representation of elements of a [Formula: see text]-module and the [Formula: see text]-theoretic version of the Chinese remainder theorem for both modules and rings.

The nice m-system of parameters for Artinian modules

This paper restates the definition of the nice m-system of parameters for Artinian modules. It also shows its effects on the differences between lengths and multiplicities of certain systems of parameters for Artinian modules: In particular, if is a nice m-system of parameters then the function is a polynomial having very nice form. Moreover, we will prove some properties of the nice m-system of parameters for Artinian modules. Especially, its effect on the annihilation of local homology modules of Artinian module A.

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

On the dual of Hilbert coefficients and width of the associated graded modules over Artinian modules

Let (A,m) be a commutative quasi-local ring with non-zero identity and let M be an Artinian co-Cohen-Macaulay R-module with NdimM = d. Let I ? m be an ideal of R with ?(0:M I) < ?. In this paper, for 0 ? i ? d, we study the dual of Hilbert coefficients ?i(I,M) of I relative to M. Also, we prove the dual of Huckaba-Marley?s inequality. Moreover, we obtain some consequences of this result.