Shadowing for families of endomorphisms of generalized group shifts

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula> be a countable monoid and let <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> be an Artinian group (resp. an Artinian module). Let <inline-formula><tex-math id="M3">\begin{document}$ \Sigma \subset A^G $\end{document}</tex-math></inline-formula> be a closed subshift which is also a subgroup (resp. a submodule) of <inline-formula><tex-math id="M4">\begin{document}$ A^G $\end{document}</tex-math></inline-formula>. Suppose that <inline-formula><tex-math id="M5">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> is a finitely generated monoid consisting of pairwise commuting cellular automata <inline-formula><tex-math id="M6">\begin{document}$ \Sigma \to \Sigma $\end{document}</tex-math></inline-formula> that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of <inline-formula><tex-math id="M7">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M8">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.</p>

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.

Reduction of Ideals Relative to an Artinian Module and the Dual of Burch’s Inequality

Let (A, 𝔪) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with ℓ(0 :M I) < ∞, then we show that for a minimal reduction J of I, (0 :M JI) = (0 :M I2) if and only if [Formula: see text] for all n ≥ 0. Moreover, we study the dual of Burch’s inequality. In particular, the Burch’s inequality becomes an equality if G(I, M) is co-Cohen-Macaulay.

On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

Some properties of Artinian modules and applications

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.

Automorphism-invariant semi-Artinian modules

It is proved that semi-Artinian module [Formula: see text] is an automorphism-invariant module if and only if [Formula: see text] is an automorphism-extendable module.

On α-almost Artinian modules

In this article we introduce and study the concept of α-almost Artinian modules. We show that each α-almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), where α ∈ {0,1}. Using this concept we extend some of the basic results of almost Artinian modules to α-almost Artinian modules. Moreover we introduce and study the concept of α-Krull modules. We observe that if M is an α-Krull module then the Krull dimension of M is either α or α + 1.

On Sequentially Co-Cohen–Macaulay Modules

In this paper, we define the notion of dimension filtration of an Artinian module and study a class of Artinian modules, called sequentially co-Cohen–Macaulay modules, which contains strictly all co-Cohen–Macaulay modules. Some characterizations of co-Cohen–Macaulayness in terms of the Matlis duality and of local homology are also given.