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Pumping lemmata are the main tool to prove that a certain language does not belong to a class of languages like the recognizable languages or the context-free languages. Essentially two pumping lemmata exist for the recognizable weighted languages: the classical one for the Boolean semiring (i.e., the unweighted case) and a similar one for fields. A joint generalization of these two pumping lemmata is provided that applies to all (right) Artinian semirings. These are (potentially non-commutative) semirings over which all finitely generated (right) semimodules have a finite bound on the length of chains of strictly increasing (right) subsemimodules. Since Artinian rings are exactly those that satisfy the Descending Chain Condition, the Artinian semirings include all skew fields and naturally also all finite semirings. The new pumping lemma thus covers most previously known pumping lemmata for recognizable weighted languages and a non-exhaustive list of semirings to which the new pumping lemma can be applied is provided as well. [ABSTRACT FROM AUTHOR]

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G (H R) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * (H R) a graph obtained from G (H R) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * (R e) denotes a subgraph of G * (H R) , induced by R e *. In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t. Also, let R be semisimple Artinian. We prove that if G * (R e) is connected for every nonzero ring component Re of R, then the connected components of G * (H R) are precisely the graphs G * (R e). [ABSTRACT FROM AUTHOR]

We study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism A → B , there is a monomorphism B → A (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, we prove that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky's dual rings) is right Artinian and right serial. Consequently, R is an Artinian PIR if and only if every factor ring of R cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. Some known results and the above one yield the following: A ring R is a direct sum of right Artinian right chain rings and Artinian PIR's if and only if every factor ring of R cogenerates its (uniform) cyclic right modules (with nonzero socle); so, such rings coincide with the right CES-rings of Jain and Lopez-Pérmouth, rings whose factors are right CF and rings that satisfy the above mentioned property for their cyclic right modules A and B. Finally, a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other. [ABSTRACT FROM AUTHOR]

In this paper, we study modules having the property that are invariant under some idempotent endomorphisms of its injective envelope. Such modules are called essentially π-injective. It is shown that (1) M is essentially π-injective iff for any essentially finite direct summand X1 of M and any submodule X2 of M with X 1 ∩ X 2 = 0 , there exists a direct summand X0 of M containing X2 such that M = X 1 ⊕ X 0 , (2) M is essentially π-injective iff M is an ef-extending right R-module and for any decomposition M = M 1 ⊕ M 2 with M1 essentially finite, M1 and M2 are relatively injective, (3) if M is essentially π-injective and R satisfies ACC on right ideals of the form r(m), m ∈ M , then M is a direct sum of uniform submodules. We also describe rings via essentially π-injective modules. It is shown that R is a semisimple artinian ring iff the direct sum of any two essentially π-injective right R-modules is essentially π-injective. [ABSTRACT FROM AUTHOR]

For each commutative ring R we associate a simple graph Γ1∗(R). We investigate the properties of an Artinian ring R when Γ1∗(R) is a star. [ABSTRACT FROM AUTHOR]

Inspired by a very recent work of A. Đurić, S. Jevđenić and N. Stopar, we introduce a new definition of zero-divisor graphs attached to rings that includes all of the classical definitions already known in the literature. We provide an interpretation of such graphs by means of a functor that we call zero-divisor functor and which is associated with a family of special equivalence relations fixed beforehand. We thus recover and generalize many known results for zero-divisor graphs and provide a framework which might be useful for further investigations on this topic. [ABSTRACT FROM AUTHOR]

We correct [3], Proposition 3.2] by showing that the class of quasi-Noetherian Lie algebras s is not E-closed. We repair the resulting gap in Example 5.1 of that paper by proving that any simple-by-soluble Lie algebra is quasi-Noetherian. More generally, any Noetherian-by-quasi-Noetherian Lie algebra is quasi-Noetherian. We also prove that any quasi-Noetherian-by-soluble Lie algebra is quasi-Noetherian, and prove E-closure for analogues of the quasi-Noetherian property for modules over associative rings and modules over Lie algebras. Using a wreath product for Lie algebras we prove that the quasi-Noetherian property is not inherited by ideals. Indeed, the derived algebra of a quasi-Noetherian Lie algebra need not be quasi-Noetherian. An analogous example is constructed for quasi-Artinian Lie algebras, which also shows that the Artinian and quasi-Artinian properties are not inherited by ideals of finite codimension. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. The prime ideal sum graph of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian nonlocal commutative rings. [ABSTRACT FROM AUTHOR]

In this work, we characterize some rings in terms of dual self-CS-Baer modules (briefly, ds-CS-Baer modules). We prove that any ring R is a left and right artinian serial ring with J2(R) = 0 iff R =M is ds-CS-Baer for every right R-module M. If R is a commutative ring, then we prove that R is an artinian serial ring iff R is perfect and every R-module is a direct sum of ds-CS-Baer R-modules. Also, we show that R is a right perfect ring iff all countably generated free right R-modules are ds-CS-Baer. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension r a (M) , where r a (M) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set Coass R H r a (M) a (M) finite, we show that r a (M) = inf { r a R p ( Hom R (R p , M)) | p ∈ Spec R } . And the class of all modules N such that Coass R N is finite is studied. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study subinjectivity domains of various R-modules and inclusion relations between these domains. We show that if the class of all subinjectivity domains is linearly ordered, then R is right Noetherian, and is either a right V-ring or a ring with unique noninjective simple module U. For the latter case, if U is projective but not indigent, then there exists a ring decomposition R=S×T such that S is a semisimple Artinian ring and T is an indecomposable right Artinian right hereditary ring. Also, in that case, if the subinjectivity domain of U is the only middle subinjectivity domain, then R is right Artinian. [ABSTRACT FROM AUTHOR]

AbstractWe prove that if f:R→S is a local homomorphism of noetherian local rings, and M is a non-zero finitely generated or artinian S-module whose injective dimension over R is bounded by the difference of the embedding dimensions of R and S, then M is an injective S-module and f is an exceptional complete intersection map. [ABSTRACT FROM AUTHOR]

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection. [ABSTRACT FROM AUTHOR]

Let φ : R → S be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between R and S are investigated. For an Artinian R-module A, the structure of A ⊗ R S is described. As an application, the Artinianess of certain local cohomology modules is clarified. [ABSTRACT FROM AUTHOR]

Let k be an arbitrary field and A be a standard graded Artinian Gorenstein k -algebra of embedding dimension four and socle degree three. Then, except for exactly one exception, A has the weak Lefschetz property. Furthermore, the exception occurs only in characteristic two. [ABSTRACT FROM AUTHOR]

Consider ideals I of the form \[ I=(x_1^2,\dots, x_n^2)+\operatorname {RLex}(x_ix_j) \] where \operatorname {RLex}(x_ix_j) is the ideal generated by all the square-free monomials which are greater than or equal to x_ix_j in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of I. Using a theorem of Lindsey [Proc. Amer. Math. Soc. 139 (2011), no. 1, 79–92], this allows for a short proof that any algebra defined by I has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran [Colloq. Math. 173 (2023), no. 1, 1–8], this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property. [ABSTRACT FROM AUTHOR]

We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semi-prorepresentable is produced. This can be seen as an analogue of Schlessinger's conditions for a functor of Artinian rings to have a semi-universal element. We also give a sufficient condition for a semi-prorepresentable formal moduli problem to admit a G-equivariant structure in a sense specified below, where G is a linearly reductive group. Finally, by making use of these criteria, we derive many classical results including the existence of (G-equivariant) formal semi-universal deformations of algebraic schemes and that of complex compact manifolds. [ABSTRACT FROM AUTHOR]

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has a radical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremum being finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J³ = 0 and without considering that rad² (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case. [ABSTRACT FROM AUTHOR]

We carry out a study of groups G in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to Z×K or Zp∞ ×K, where K is a finite group and p is a prime number. We also prove that a group G is infinitely generated restricted-finite if and only if G = AT where A and T are subgroups of G such that A is normal quasi-cyclic and T is finite. As an application of our results, we show that if G is not torsion with finite G′ and the group-ring RG has restricted minimum condition, then R is a semisimple ring and G ∼= T ⋊ Z, where T is finite whose order is unit in R. The converse is also true with certain conditions including G = T × Z. [ABSTRACT FROM AUTHOR]

There are different mechanisms in the literature to measure how flat a module can be. We study an alternative perspective on the analysis of the flatness of a module, as we assign to every module a class of short exact sequences of modules, namely flatly generated proper classes. We focus on modules that generate flat proper classes, aiming for them to be as small as possible. We refer to such modules as being τ-rugged, as opposed to flat modules. Properties of τ-rugged modules are studied. We study the structure of rings whose certain types of modules are either flat or τ-rugged. Specifically, we prove that if R is a right Noetherian ring, then every (finitely presented) nonflat right R-module is τ-rugged if and only if R has a unique (up to isomorphism) singular simple right R-module and it is either Artinian serial ring with J2(R) = 0 or right finitely ∑-CS, right SI ring. If R is a right perfect ring with nonflat finitely presented simple right R-module U, then U is τ-rugged if and only if every nonflat right R-module is τ-rugged if and only if R has a unique (up to isomorphism) singular simple right R-module U and it is Artinian serial ring with J2(R) = 0. In addition, if R is commutative and every nonflat R-module is τ-rugged, then R is either a von Neumann regular ring or an fp-injective ring. [ABSTRACT FROM AUTHOR]

We develop a formalism of unit |$F$| -modules in the style of Lyubeznik and Emerton-Kisin for rings that have finite |$F$| -representation type after localization and completion at every prime ideal. As applications, we show that if |$R$| is such a ring then the iterated local cohomology modules |$H^{n_{1}}_{I_{1}} \circ \cdots \circ H^{n_{s}}_{I_{s}}(R)$| have finitely many associated primes, and that all local cohomology modules |$H^{n}_{I}(R / gR)$| have closed support when |$g$| is a nonzerodivisor on |$R$|⁠. [ABSTRACT FROM AUTHOR]

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph E A G (R) associated to a commutative ring R , denoted by dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) , respectively. Here we prove the finiteness conditions of dim M (E A G (R)) and dim + (E A G (R)). In addition, we characterize dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) for artinian rings and the direct product of rings. [ABSTRACT FROM AUTHOR]

Let n be an even natural number. We compute the periods of any n 2 -dimensional complete intersection algebraic cycle inside an n -dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting. [ABSTRACT FROM AUTHOR]

Let R be a ring and J(R), C(R) its Jacobson radical and center. If R is a centrally essential (CE ring for brevity) ring with R / J (R) commutative, then any minimal right ideal is contained in C(R). In a CE ring, every minimal right or left ideal is an ideal. A right Artinian (subdirectly indecomposable (SI, for brevity) right Noetherian) CE ring is an Artinian (a local Artinian) ring. R is duo if all factor rings of R are CE. We describe Noetherian SI CE rings and CE rings with SI center. We give examples of non-commutative SI CE rings. [ABSTRACT FROM AUTHOR]

Recall that a ring R is a Hilbert ring if any maximal ideal of R [ X ] contracts to a maximal ideal of R. The main purpose of this paper is to characterize the prime ideals of a commutative ring R which are traces of the maximal ideals of the polynomial ring R [ X ]. In this context, we prove that if p is a prime ideal of R such that R / p is a semi-local domain of (Krull) dimension ≤ 1 , then p is the trace of a maximal ideal of R [ X ]. Whereas, if R is Noetherian and either (dim (R / p) ≥ 2) or (the quotient field of R / p is algebraically closed, dim (R / p) = 1 and R / p is not semi-local), then p is never the trace of a maximal ideal of R [ X ]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings R [ X ] and the finiteness of the Ext-index of localizations of R. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures. [ABSTRACT FROM AUTHOR]

Let (, ,) be an Ext -finite, Krull–Schmidt and k -linear n -exangulated category with k a commutative artinian ring. In this note, we prove that has Auslander–Reiten–Serre duality if and only if has Auslander–Reiten n -exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander–Reiten–Serre duality). Finally, assume further that has Auslander–Reiten–Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander–Reiten–Serre duality. [ABSTRACT FROM AUTHOR]

In this paper, we study σ -skew McCoy rings under the σ -compatible or the σ -semicompatible conditions. We show that if R is a semicommutative right or left artinian ring which is σ -semicompatible with an epimorphism σ , then the Jacobson radical J (R) is σ -skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every σ -compatible right duo ring is σ -skew McCoy and that for σ -compatible regular rings, the notions of the σ -skew McCoy and the right McCoy coincide. In addition, we show that every σ -semicompatible semicommutative ring is linearly σ -skew Camillo and that every matrix ring over a division ring is linearly σ -skew Camillo for any endomorphism σ. [ABSTRACT FROM AUTHOR]

An R-module M is called almost uniserial if any two non-isomorphic submodules of M are comparable. A ring R is an almost left uniserial ring if RR is almost uniserial. In this paper, we introduce a class of artinian almost uniserial rings. Also we give a classification of almost uniserial modules over principal ideal domains. [ABSTRACT FROM AUTHOR]

We prove that in the polynomial ring Q = k [ x , y , z , w ] , with k an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals I satisfying the inclusions (x , y , z , w) 4 ⊆ I ⊆ (x , y , z , w) 2 can be obtained by doubling from a grade three perfect ideal J ⊂ I such that Q / J is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the Q -module Q / I can be completely described in terms of a graded minimal free resolution of the Q -module Q / J and a homogeneous embedding of a shift of the canonical module ω Q / J into Q / J. [ABSTRACT FROM AUTHOR]

This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K-algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite Q -algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the Q -algebra Q [ V n ] of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set [ n ] : = { 1 , 2 , ... , n } . All results are equally true if we replace Q [ V n ] by K [ V n ] , where K is any perfect field of characteristics ≠ 2 . The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework. [ABSTRACT FROM AUTHOR]

A ring R is projective invariant Baer (π -Baer for short) if the right annihilator of every projection invariant left ideal of R is generated by an idempotent element of R. A ring R is called weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left s -unital by left semicentral idempotents, which implies that R modulo, the right annihilator of any principal right ideal, is flat. We study relations between the π -Baer and weakly p.q.-Baer properties of a ring R , and its skew Hurwitz series ring R H [ [ x ; σ ] ] , where R is a ring equipped with an endomorphism σ. Examples are provided to explain the results. [ABSTRACT FROM AUTHOR]

We study homological behavior of modules satisfying the Auslander condition. Assume that AC is the class of left R -modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in AC belongs to AC for any ring R. As a consequence, we show that for any left Noetherian ring R , AC is a resolving subcategory of the category of left R -modules if and only if R R satisfies the Auslander condition if and only if each Gorenstein projective left R -module belongs to AC. As an application, we prove that, for an Artinian algebra R satisfying the Auslander condition, R is Gorenstein if and only if AC coincides with the class of Gorenstein projective left R -modules if and only if (AC < ∞ , (AC < ∞) ⊥) is a tilting-like cotorsion pair if and only if (AC < ∞ , I) is a tilting-like cotorsion pair, where AC < ∞ is the class of left R -modules with finite AC -dimension and I is the class of injective left R -modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein. [ABSTRACT FROM AUTHOR]

The set is a sub-f-ring of RL, that is, the ring of all continuous real-valued functions on a completely regular frame L. The main purpose of this paper is to continue our investigation begun in [3] of extending ring-theoretic properties in RL to the context of completely regular frames by replacing the ring RL with the ring Cc(L) to the context of zero-dimensional frames. We show that a frame L is a CP-frame if and only if Cc(L) is a regular ring if and only if every ideal of Cc(L) is pure if and only if Cc(L) is an Artin-Rees ring if and only if every ideal of Cc(L) with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring Cc(L)=hi is an Artin-Rees ring for any 2 Cc(L) if and only if every minimal prime ideal of Cc(L) is an Artin-Rees ideal. [ABSTRACT FROM AUTHOR]

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

Recall that an R-module M is pure-semisimple if every module in the category σ [ M ] is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that "a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring". Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is "whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?" The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals M = R x ⊕ T which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple. [ABSTRACT FROM AUTHOR]

We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring \Gamma _e ≔(1-e)R(1-e) of a semiperfect noetherian basic ring R by a construction inside \mathsf {mod}\,R. This is then applied to investigate homological properties of idempotent subrings \Gamma _e under the assumption of R/\langle 1-e\rangle being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module S_e ≔eR /\operatorname {rad}eR with \operatorname {Ext}_R^1(S_e,S_e) = 0 is self-orthogonal, that is \operatorname {Ext}^k_R(S_e,S_e) vanishes for all k \geq 1, whenever \operatorname {gldim}R and \operatorname {pdim}eR(1-e)_{\Gamma _e} are finite. Indeed, a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose e \in R is an idempotent such that all idempotent subrings \Gamma sandwiched between \Gamma _e and R, that is \Gamma _e \subseteq \Gamma \subseteq R, have finite global dimension. Then the simple summands of S_e can be numbered S_1, \dots, S_n such that \operatorname {Ext}_R^k(S_i, S_j) = 0 for 1 \leq j \leq i \leq n and all k > 0. [ABSTRACT FROM AUTHOR]

Let k be commutative artinian ring and (C , E , s) be an Ext-finite, Krull-Schmidt n-exangulated category. We define two additive subcategories C r and C l of C in terms of the representation functors from the stable category of C to k-modules. Moreover, we introduce two quasi-inverse functors τ n : C r ¯ → C l ¯ and τ n − : C l ¯ → C r ¯ . Furthermore, we investigate the subcategories C r and C l in terms of morphisms determined by objects, and then give equivalent characterizations for the existence of Auslander-Reiten n-exangles. [ABSTRACT FROM AUTHOR]

Let be an ideal of a commutative noetherian ring R and M an R -module with Cosupport in V (). We show that M is -coartinian if and only if Ext R i (R / , M) is artinian for all 0 ≤ i ≤ cd (, M) , which provides finite steps to examine -coartinianess. We also consider the duality of Hartshorne's questions: for which rings R and ideals are the modules H i (M) -coartinian for every artinian R -module M and all i ≥ 0 ; whether the category (R ,) coa of -coartinian modules is an abelian subcategory of the category of R -modules, and establish affirmative answers to these questions in the case cd (, R) ≤ 1 and dim R / ≤ 1. [ABSTRACT FROM AUTHOR]

We study rings R such that all (or certain classes) of right R-modules satisfy chain conditions on essential endosubmodules. It is proved that if all right R-modules have DCC on essential endosubmodules, then R is a right pure semisimple ring, and if all right R-modules have both ACC and DCC on essential endosubmodules, then R is of finite representation type. Rings R such that all right R-modules have ACC on essential endosubmodules are discussed, and it is shown, in particular, that if such a ring R is right Artinian, then R is left pure semisimple. [ABSTRACT FROM AUTHOR]

Let A be a ring with minimum condition on principal right ideals. It is proved that countably distributive right (left) A-modules coincide with Artinian (Noetherian) right (left) A-modules. Rings, over which all right modules are ℵ 0 -distributive coincide with rings of finite representation type. Rings, whose right modules are semidistributive, coincide with Kawada rings, over basis rings of which all right modules are completely cyclic. [ABSTRACT FROM AUTHOR]

A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l , where each D i is a principal ideal domain and each A i is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 픭 , and R / 픭 is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided. [ABSTRACT FROM AUTHOR]

Let R be an Artinian ring with a dualizing complex D•. Then (1) if the injective modules generate the derived category D (R) as a triangulated category with arbitrary coproducts, then the finitistic dimension of R is finite; (2) if the injective modules generate the homotopy category K (R − lnj) as a triangulated category with arbitrary coproducts, then the global dimension of R is finite. The first result generalizes Theorem 4.3 in [Adv. Math., 2019, 354: 106735, 21 pp.]. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we provide a fairly complete description of a simple ring R with involution f such that relative to f the traces of all elements of R form a commutative subset of R. This description is based on the characteristic of R. While the case of characteristic not 2 readily follows from current results in the literature, by contrast, the opposite case of characteristic 2 requires markedly more work. In the rest of the paper, the results carried out earlier are put to work to delineate the structure of a simple Artinian ring R with involution f, which is equipped with a given non central multiplicative subset M stabilized by f and preserved by all inner automorphisms of R, and such that the traces relative to f of all elements of M form a commutative subset of R. [ABSTRACT FROM AUTHOR]

Proper classes (or exact structures) offer rich research topics due to their important role in category theory. Motivated by the studies on opposite of injective modules, we introduce a new approach to opposed to injectivity in terms of injectively generated proper classes. The smallest possible proper class generated injectively by a single module is the class of all split short exact sequences. We call a module M ι-indigent if the proper class injectively generated by M consists only of split short exact sequences. We are able to show that if R is a ring which is not von Neumann regular, then every right (pure-injective) R-module is either injective or ι-indigent if and only if R is an Artinian serial ring with J2 (R) = 0 and has a unique non-injective simple right R-module up to isomorphism. Moreover, if R is a ring such that every simple right R-module is pure-injective, then every simple right R-module is ι-indigent or injective if and only if R is either a right V -ring or R = A × B, where A is semisimple, and B is an Artinian serial ring with J2 (B) = 0. We investigate the class ι(R) which consists of those proper classes such that is injectively generated by a module. We call such a class (right) proper injective profile of a ring R. We prove that if R is an Artinian serial ring with J2 (R) = 0, then |ι(R)| = 2n, where n is the number of non-isomorphic non-injective simple right R-modules. In addition, if ι(R) is a chain, then R is a right Noetherian ring over which every right R-module is either projective or i-test, and has a unique singular simple right R-module. Furthermore, in this case, R is either right hereditary or right Kasch. We observe that |ι(R)| ≠ 3 for any ring R which is not von Neumann regular. We construct a bounded complete lattice structure on ι(R) in case ι(R) is a partially ordered set under set inclusion. Moreover, if R is an Artinian serial ring with J2 (R) = 0, then this lattice structure is Boolean. [ABSTRACT FROM AUTHOR]

Let (C , E , s) be an Ext-finite, Krull-Schmidt and k-linear n-exangulated category with k a commutative artinian ring. In this note, we define two additive subcategories C r and C l of C in terms of the representable functors from the stable category of C to the category of finitely generated k-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten n-exangles via determined morphisms. These results unify and extend their works by Jiao–Le for exact categories, Zhao–Tan–Huang for extriangulated categories, Xie–Liu–Yang for n-abelian categories. [ABSTRACT FROM AUTHOR]

Codimension two Artinian algebras have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A , we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees. [ABSTRACT FROM AUTHOR]

In this paper, we say a ring R is Nil*-Artinian if any descending chain of nil ideals stabilizes. We first study Nil*-Artinian properties in terms of quotients, localizations, polynomial extensions and idealizations, and then study the transfer of Nil* -Artinian rings to amalgamated algebras. Besides, some examples are given to distinguish Nil* -Artinian rings, Nil* -Noetherian rings and Nil* -coherent rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of μ-weakly Hopfian modules which is a new generalization of Hopfian modules. It is shown that over a ring R, every quasi-projective (projective, free) R-module is μ-weakly Hopfian if and only if R has no nonzero semisimple injective R-module. Some basic characterizations of projective μ-weakly Hopfian modules are proved. We demonstrate that if the ACC holds on μ-small submodules of an R-module M, then M is μ-weakly Hopfian. Other properties of μ-weakly Hopfian modules are also obtained with examples. [ABSTRACT FROM AUTHOR]

Let R be a ring. The class of SA-injective right R-modules (SAIR) is introduced as a class of soc-injective right R-modules. Let N be a right R-module. A right R-module M is said to be SA-N-injective if every R-homomorphism from a semi-artinian submodule of N into M extends to N. A module M is called SA-injective, if M is SA-R-injective. We characterize rings over which every right module is SA-injective. Conditions under which the class SAIR is closed under quotient (resp. dIRect sums, pure homomorphic images) are given. The definability of the class SAIR is studied. Finally, relations between SA-injectivity and certain generalizations of injectivity are given. [ABSTRACT FROM AUTHOR]