Alberto Facchini

• Professor (Full) at University of Padua

We describe the endomorphism ring of a short exact sequences $0 \to A_R \to B_R \to C_R \to 0$ with $A_R$ and $C_R$ uniserial modules and the behavior of these short exact sequences as far as their direct sums are concerned.

We study the category $\operatorname{Morph}(\operatorname{Mod} R)$ whose objects are all morphisms between two right $R$-modules. The behavior of objects of $\operatorname{Morph}(\operatorname{Mod} R)$ whose endomorphism ring in $\operatorname{Morph}(\operatorname{Mod} R)$ is semilocal is very similar to the behavior of modules with a semilocal end...

We investigate the relation between the notion of $e$-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor $P\colon\operatorname{Mod} R\to \operatorname{Spec}(\operatorname{Mod} R)$, where $\operatorname{Spec}(\operatorname{Mod} R)$ denotes the spectral category of $\operatorname{Mod} R$...

We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $\mathbb N...

In this first chapter, the basic notions about rings, modules and categories are given. For instance we give the notion of ring morphism, module morphism, functor, we describe the difference between right modules and left modules, we define bimodules, submodules and quotients. We introduce natural transformations of functors. We study direct sums a...

In this chapter, we introduce and study some of the most important classes of modules: free modules, projective modules, simple modules, semisimple modules, noetherian modules, artinian modules, modules of finite composition length. During the study of these seven classes of modules, we have the opportunity of touching several fundamental topics in...

In this chapter we have collected some further notions about categories, preadditive categories, additive categories and abelian categories. Thus we present monomorphisms in arbitrary categories, epimorphisms, subobjects, quotient objects, products and coproducts of objects, biproducts in a preadditive category, initial objects, null objects, termi...

The aim of this paper is to propose the study of a class of Lie-admissible algebras. It is the class (variety) of all the (not-necessarily associative) algebras $M$ over a commutative ring $k$ with identity $1_k$ for which $(x,y,z)=(y,x,z)+(z,y,x)$ for every $x,y,z\in M$. Here $(x,y,z)$ denotes the associator of $M$. We call such algebras {\em alge...

We show that, making use of multiplicative lattices and idempotent endomorphisms of an algebraic structure A, it is possible to derive several notions concerning A in a natural way. The multiplicative lattice necessary here is the complete lattice of congruences of A with multiplication given by commutator of congruences. Our main application is to...

We study commutators of congruences, idempotent endomorphisms and semidirect-product decompositions of heaps and trusses.

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counter...

We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of $L$-algebras nor that o...

In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring $k$ with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan alge...

We study the notions of action, semidirect product and commutator of ideals for digroups and skew braces.

We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible $k$-algebras. Associating with any algebra $(A,\cdot)$ its sub-adjacent anticommutative algebra $(A,[-,-])$ i...

General results on multiplicative lattices found recently by Facchini, Finocchiaro and Janelidze have been studied in the particular case of groups by Facchini, de Giovanni and Trombetti. In this paper we prove that these results hold not only for the multiplicative lattices of all normal subgroups of a group, but also for much more general multipl...

We show that there is a little mistake in an implication in a paper of Bob Gilmer on rngs.

We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, “product” of ideals of skew braces is often considered. We show that the so-called (Huq = Smith) condition holds for left skew braces. Finally, we give a set of generators for the com...

The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is a splitting monomorphism in the category o...

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counter...

The aim of this paper is to investigate the behaviour of prime and semiprime subgroups of groups, and their relation with the existence of abelian normal subgroups. In particular, we study the set Spec( G ) of all prime subgroups of a group G endowed with the Zariski topology and, among other examples, we construct an infinite group whose proper no...

We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove t...

We describe the structure of projective covers of modules over a local ring, when such covers exist, and modules with minimal sets of generators. The case of modules over valuation rings is studied in more detail.

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum \({\mathrm {Spec}}(R)\) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ide...

We continue our study of the relation between existence of projective covers and existence of minimal sets of generators for right modules over an arbitrary ring.

General results on multiplicative lattices found recently by Facchini, Finocchiaro and Janelidze have been studied in the particular case of groups by Facchini, de Giovanni and Trombetti. In this paper we prove that these results hold not only for the multiplicative lattices of all normal subgroups of a group, but also for much more general multipl...

These are the notes of a non-standard course of Algebra. It deals with elementary theory of commutative monoids and non-commutative rings. Most of what is taught in a master course of Commutative Algebra holds not only for commutative rings, but more generally for any commutative monoid, which shows that the additive group structure on a commutativ...

We show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak formof theKrull–Schmidt theorem holds for direct sums of these ker...

We consider a small category naturally associated with any fixed R-S-bimodule MSR. The class of objects of this category is the underlying set M of MSR. Some additive decompositions of the elements of the bimodule MSR appear naturally. They are the analog of the usual decompositions of the identity 1R of a ring R as sums of pairwise orthogonal idem...

A ringed partially ordered set with zero is a pair (L,F) , where L is a partially ordered set with a least element 0_L and F\colon L\to\mathsf{Ring} is a covariant functor. Here the partially ordered set L is given a category structure in the usual way and \mathsf{Ring} denotes the category of associative rings with identity. Let \mathsf{RingedParO...

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum $\mathrm{Spec}(R)$ of a unital commutative ring $R$ is always a spectral (=coherent) topological space. In this generalization, which includes several other known ones, the role of ideals...

We describe the structure of the projective cover of a module $$M_R$$ M R over a local ring R and its relation with minimal sets of generators of $$M_R$$ M R . The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect.

We apply minimal weakly generating sets to study the existence of Add(UR)-covers for a uniserial module UR. If UR is a uniserial right module over a ring R, then S:=End(UR) has at most two maximal (right, left, two-sided) ideals: one is the set I of all endomorphisms that are not injective, and the other is the set K of all endomorphisms of UR that...

This book provides an introduction to some key subjects in algebra and topology. It consists of comprehensive texts of some hours courses on the preliminaries for several advanced theories in (categorical) algebra and topology. Often, this kind of presentations is not so easy to find in the literature, where one begins articles by assuming a lot of...

The paper is devoted to a kind of “very non-abelian” spectral categories. Under strong conditions on a category \mathcal X , we prove, among other things, that, for a given faithful localization \mathcal C \to \mathcal X , we have canonical equivalences Spec (\mathcal{C})\sim\mathcal{X}\sim (category of injective objects in \mathcal{C}) , and that...

We study the category Morph(Mod-R) whose objects are all morphisms between two right R-modules. The behavior of the objects of Mod-R whose endomorphism ring in Morph(Mod-R) is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum ⊕i=1nMi, that is, block-diago...

We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class \( \mathcal {SF}\) of strongly flat modules when R is a right Ore domain with classical right quotient...

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair (T, F) of full replete subcategories in a category C, the corresponding full subcategory Z=T∩F of trivial objects in C. The morphisms which factor through Z are called Z-trivial, and these form an ideal of morphisms, with respect to...

For a category with finite limits and a class of monomorphisms in that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable -essential monomorphisms in to construct a spectral category . We show that it has finite limits and that the canonical functor prese...

The paper is devoted to a kind of `very non-abelian' spectral categories. Under strong conditions on a category $\mathcal{X}$, we prove, among other things, that, for a given faithful localization $\mathcal{C}\to\mathcal{X}$, we have canonical equivalences $\mathrm{Spec}(\mathcal{C})\sim\mathcal{X}\sim(\mathrm{Category\,\,of\,\,injective\,\,objects...

We apply minimal weakly generating sets to study the existence of Add$(U_R)$-covers for a uniserial module $U_R$. If $U_R$ is a uniserial right module over a ring $R$, then $S:=$End$ (U_R)$ has at most two maximal (right, left, two-sided) ideals: one is the set $I$ of all endomorphisms that are not injective, and the other is the set $K $ of all en...

Occasioned by the international conference "Rings and Factorizations" held in February 2018 at University of Graz, Austria, this volume represents a wide range of research trends in the theory of commutative and non-commutative rings and their modules, including multiplicative ideal theory, Dedekind and Krull rings and their generalizations, rings...

Factorizations of ring elements are described by finite chains of principal ideals. We use the description of cyclically presented modules over local rings to study factorization of elements in local rings.

The aim of this chapter is to point out some aspects of additive categories that usually are not sufficiently stressed in a first course in category theory.

Now we will present a construction due to [Gabriel and Oberst]. Recall that a Grothendieck category is an abelian category with arbitrary coproducts, with exact direct limits and with a generator. Every Grothendieck category is a category with injective envelopes (see, for instance, [Popescu, Theorem 3.10.10]).

The idea of the Auslander–Bridger transpose is the following. We have a finitely presented right R-module MR, defined by m generators and s relations say. Thus MR has a free presentation

In this section, we recall the notions of Goldie dimension of a modular lattice and Goldie dimension of a module. In fact, the notion of Goldie dimension of a module concerns the lattice L(M) of all submodules of M, which is a modular lattice.

In this chapter, we review what we will need in the rest of the book as far as commutative monoids are concerned. This will show how much we assume of the reader. The contents of Sections 1.5 and 1.7 are exceptions: they are completely independent of the rest of the chapter. In Section 1.5, some in-depth examinations of the theory of Krull monoids...

We will now study the maximal ideals of a preadditive category C. We say that an ideal M of a preadditive category C is maximal if the improper ideal HomC of C is the unique ideal of the category C properly containing M. Obviously, if all objects of C are zero objects, maximal ideals do not exist in C. In this section, we will characterize maximal...

In the first part of this chapter, we will describe what occurs if in the Krull– Schmidt–Azumaya theorem (Theorem 4.34), instead of considering modules with local endomorphism rings, that is, endomorphism rings with one maximal ideal, we consider modules whose endomorphism rings have two maximal ideals.

small if for every family {Mi | i ∈ I } of right R-modules and every homomorphism ϕ: MR → ⊕i∈IMi, there is a finite subset F ⊆ I such that ϕ(M) ⊆ ⊕i∈FMi.

In this final chapter, we pose some questions and present some open problems. Some of them are classical problems.

All rings we will consider will be associative rings with identity, and all modules will be unital modules. They will be right modules unless otherwise specified. Since we want all modules MR to have an endomorphism ring End(MR) (the zero module as well), the identity of the ring may be equal to zero, in which case the ring trivially reduces to zer...

A completely prime ideal P of a ring S is a proper ideal P of S with the property that for every x, y ∈ S, xy ∈ P implies that either x ∈ P or y ∈ P. Notice that if ϕ: S → D is a ring morphism of S into an integral domain D (e.g., a division ring D), then the kernel of ϕ is a two-sided completely prime ideal of S.

We introduced and defined rings and modules of finite type in Section 8.1, but then the rest of Chapter 8 was devoted to studying modules of type ≤ 2. (Auslander– Bridger modules and dual Auslander–Bridger modules do not have type ≤ 2 in general, but they do have semilocal endomorphism rings.) In this chapter we will see some further results on mod...

We study the structure of the commutative multiplicative monoid \(\mathbb {N}_0[x]^*\) of all the non-zero polynomials in \(\mathbb {Z}[x]\) with non-negative coefficients. The monoid \(\mathbb {N}_0[x]^*\) is not half-factorial and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into \(\mathbb...

We show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and...

Let $\mathsf{PreOrd}(\mathbb C)$ be the category of internal preorders in an exact category $\mathbb C$. We show that the pair $(\mathsf{Eq}(\mathbb C), \mathsf{ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$, where $\mathsf{Eq}(\mathbb C)$ and $\mathsf{ParOrd}(\mathbb C)$) are the full subcategories of internal equivale...

Several elementary properties of the symmetric group Sn extend in a nice way to the full transformation monoid Mn of all maps of the set X := {1, 2, 3,…,n} into itself. The group Sn turns out to be the torsion part of the monoid Mn. That is, there is a pretorsion theory in the category of all maps f:X → X, X an arbitrary finite set, in which biject...

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair $(\mathcal T, \mathcal F)$ of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of trivial objects in $\mathcal C$. The morphisms which factor through...

We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦Hom(R) turns out to be a contravariant functor of the category Ring of associative rings with i...

For a right module MR over a ring R, we consider the set I of all the endomorphisms φ ∈ E := End(MR) that are not injective and the set K of all the endomorphisms φ that are not surjective. We prove that when MR is a uniserial module, then E/K is a left chain domain and E/I is a right chain domain. The technique we make use of to prove this can be...

For a category $\mathcal{C}$ with finite limits and a class $\mathcal{S}$ of monomorphisms in $\mathcal{C}$ that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable $\mathcal{S}$-essential monomorphisms in $\mathcal{C}$ to construct a spectral category $\m...

Several elementary properties of the symmetric group $S_n$ extend in a nice way to the full transformation monoid $M_n$ of all maps of the set $X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some sense the torsion part of the monoid $M_n$. More precisely, there is a pretorsion theory in the category of all maps $f\colon X\to...

We show that in the category of preordered sets, there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion an...

We present the basic results concerning isoartinian and isonoetherian rings and modules. These rings and modules, which were essentially defined in two papers of ours recently published in the {\em Journal of Algebra}, generalize artinian and noetherian rings and modules. In particular, here we focus on the isoradical of a ring, which is a generali...

This book collects and coherently presents the research that has been undertaken since the author’s previous book Module Theory (1998). In addition to some of the key results since 1995, it also discusses the development of much of the supporting material. In the twenty years following the publication of the Camps-Dicks theorem, the work of Facchin...

We associate to any ring $R$ with identity a partially ordered set Hom$(R)$, whose elements are all pairs $(\mathfrak a,M)$, where $\mathfrak a=\ker\varphi$ and $M=\varphi^{-1}(U(S))$ for some ring morphism $\varphi$ of $R$ into an arbitrary ring $S$. Here $U(S)$ denotes the group of units of $S$. The assignment $R\mapsto{}$Hom$(R)$ turns out to be...

We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring $R$ that coincides with the $R$-topology defined by Matlis when $R$ is commutative. (2) We consider the class $ \mathcal{SF}$ of strongly flat modules when $R$ is a right Ore domain with classical right quot...

We study the existence of maximal ideals in preadditive categories defining an order $\preceq$ between objects, in such a way that if there do not exist maximal objects with respect to $\preceq$, then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an e...

In this article, we present the classical Krull-Schmidt Theorem for groups, its statement for modules due to Azumaya, and much more modern variations on the theme, like the so-called weak Krull-Schmidt Theorem, which holds for some particular classes of modules. Also, direct product of modules is considered. We present some properties of the catego...

https://www.tandfonline.com/eprint/73gPS8NMN5r4dgRBywb3/full

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups 𝓝(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemar...

In a Dedekind domain $D$, every non-zero proper ideal $A$ factors as a product $A=P_1^{t_1}\cdots P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain $D$, the $D$-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal $A$ of an arbitrary ring $R$ as a pr...

In a Dedekind domain $D$, every non-zero proper ideal $A$ factors as a product $A=P_1^{t_1}\cdots P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain $D$, the $D$-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal $A$ of an arbitrary ring $R$ as a pr...

We describe the endomorphism ring of a short exact sequence 0 →AR →BR →CR →0 with AR and CR uniserial modules and the behavior of these short exact sequences as far as their direct sums are concerned.

In dealing with monoids, the natural notion of kernel of a monoid morphism \(f:M\rightarrow N\) between two monoids M and N is that of the congruence \(\sim _f\) on M defined, for every \(m,m'\in M\), by \(m\sim _fm'\) if \(f(m)=f(m')\). In this paper, we study kernels and equalizers of monoid morphisms in the categorical sense. We consider the cas...

We study the existence of maximal ideals in preadditive categories defining an order $\preceq$ between objects, in such a way that if there do not exist maximal objects with respect to $\preceq$, then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an e...

Let $R$ be a right and left Ore ring, $S$ its set of regular elements and $Q = R[S^{-1}] = [S^{-1}] R$ the classical ring of quotients of $R$. We prove that if F.dim$(Q_Q) = 0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $ (ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii) $ $h$-...

Let $R$ be a right and left Ore ring, $S$ its set of regular elements and $Q = R[S^{-1}] = [S^{-1}] R$ the classical ring of quotients of $R$. We prove that if F.dim$(Q_Q) = 0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $ (ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii) $ $h$-...

https://authors.elsevier.com/a/1UyJy4~FOuwX2

The first author was partially supported by Università di Padova (Progetto ex 60% “Anelli e categorie di moduli”) and Fondazione Cassa di Risparmio di Padova e Rovigo (Progetto di Eccellenza “Algebraic structures and their applications”.) The second author was partially supported by a grant from IPM (No. 95160078).

Let G be a group. We analyse some aspects of the category G-Grp of G-groups. In particular, we show that a construction similar to the construction of the spectral category, due to P. Gabriel and U. Oberst, and its dual, due to the second author, is possible for the category G-Grp.

We study modules with chain conditions up to isomorphism, in the following sense. We say that a right module M is isoartinian if, for every descending chain M≥M1≥M2≥. . . of submodules of M, there exists an index n≥1 such that Mn is isomorphic to Mi for every i≥n. A ring R is right isoartinian if RR is an isoartinian module. Similarly we define iso...

We study the relations between product decomposition of singular matrices into products of idempotent matrices and product decomposition of invertible matrices into elementary ones.

We extend the classical theory of factorization in noncommutative integral domains to the more general classes of right saturated rings and right cyclically complete rings. Our attention is focused, in particular, on the factorizations of right regular elements into left irreducible elements. We study the connections among such factorizations, righ...

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [55. Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 . View all references]. In this paper, we show that this is not necesarily true for modules over noncommutative rings...

A module M is called automorphism-invariant if it is invariant under automorphisms of its injective envelope. In this paper, we study the en-domorphism rings of automorphism-invariant modules and their injective envelopes. We investigate some cases where automorphism-invariant modules are quasi-injective and a connection between automorphism-invari...

We study noncommutative continuant polynomials via a new leapfrog construction. This needs the introduction of new indeterminates and leads to generalizations of Fibonacci polynomials, Lucas polynomials and other families of polynomials. We relate these polynomials to various topics such as quiver algebras and tilings. Finally , we use permanents t...

We consider some generalizations of the notion of divisible module, and study when our “divisible” modules coincide with injective modules. Here the base ring \(R\) is usually a (not-necessarily commutative) integral domain. For instance, we prove that an integral domain \(R\) is right noetherian and right hereditary if and only if the class of inj...

There is a curious connection between decimal representations of rational numbers, the structure of finite cyclic monoids, divisibility rules between integers, and divisors of the numbers of the form 99 . . . 900 . . . 0. In all of these cases, we find not only periodicity from some point on, but also the same type of periodicity.