Search

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]

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]

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]

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]

In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C3 properties and M/SocM has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak FI-extending, pseudo duo, C3 properties and ascending (or descending) chain condition on essential submodules then M = M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (or Artinian, respectively) submodule M2. Moreover, we show that a nonsingular weak CS (or weak C* 11, or weak FI) module has a direct summand which essentially contains the socle of the module and is a CS (or C11, or FI-extending, respectively) module. [ABSTRACT FROM AUTHOR]

In this paper, we relate some properties of non-comaximal graph of ideals of a commutative ring with identity with the properties of the ring. [ABSTRACT FROM AUTHOR]

We answer an open problem posed by Iarrobino in the '80s: is there an elementary component of the Hilbert scheme of points Hilbd(An) with dimension less than (n-1)(d-1)? We construct an infinite class of such components in Hilbd(A4). Our techniques also allow us to construct an explicit example of a local Artinian ring with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension 2. [ABSTRACT FROM AUTHOR]

This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique. [ABSTRACT FROM AUTHOR]

A local ring is an associative ring with unique maximal ideal. We associate with each Artinian local ring with singleton basis four invariants (positive integers) p , n , s , t. The purpose of this article is to describe the structure of such rings and classify them (up to isomorphism) with the same invariants. Every local ring with singleton basis can be constructed over its coefficient subring by a certain polynomial called the associated polynomial. These polynomials play significant role in the enumeration. [ABSTRACT FROM AUTHOR]

In this paper, we determine the structure of rings that have secondary representation (called representable rings) and give some characterizations of these rings. Also, we characterize Artinian rings in terms of representable rings. Then we introduce completely representable modules, modules every non-zero submodule of which have secondary representation, and give some necessary and sufficient conditions for a module to be completely representable. Finally, we define strongly representable modules and give some conditions under which representable module is strongly representable. [ABSTRACT FROM AUTHOR]

This paper proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator L ∈ D for an element m in a D -module M. The main idea in this paper is to look for submodules of M. If N is a non-trivial submodule of M , constructing the minimal annihilator R of the image of m in M / N gives a right-factor of L in D. Then L = L ′ R where the left-factor L ′ is the telescoper of R (m) ∈ N. To expedite computing L ′ , compute the action of D on a natural basis of N , then obtain L ′ with a cyclic vector computation. The next main idea is to construct submodules from automorphisms, if we can find some. An automorphism with distinct eigenvalues can be used to decompose N as a direct sum N 1 ⊕ ⋯ ⊕ N k. Then L ′ is the LCLM (Least Common Left Multiple) of L 1 , ... , L k where L i is the telescoper of the projection of R (m) on N i. An LCLM can greatly increase the degrees of coefficients, so L ′ and L can be much larger expressions than the factors L 1 , ... , L k and R. Examples show that computing each factor L i and R separately can save a lot of CPU time compared to computing L in expanded form with standard creative telescoping. [ABSTRACT FROM AUTHOR]

We provide criteria for identifying exact pairs of zero-divisors from zero-divisor graphs of commutative rings, and extend these criteria to compressed zero-divisor graphs. Finally, our results are translated as constructions for exact zero-divisor subgraphs. [ABSTRACT FROM AUTHOR]

In this paper, we show that if M is a non-zero Artinian R-module and x := x1;...; xn is anM-coregular sequence, then x1;...; xn is a D(Hx n(M))-coregular sequence. Moreover, if R is complete with respect to I-adic topology and d = NdimM, then dimHI d (M) = d and depthHd I (M) = minf2; dg whenever HI d (M) 6= 0:. [ABSTRACT FROM AUTHOR]

A moduleM is called ss-lifting if for every submodule A of M, there is a decomposition M = M1 ⊕ M2 such that M1 ≤ A and A∩M2 ⊆ Socs (M), where Socs(M) = Soc(M)∩Rad(M). In this paper, we provide the basic properties of ss-lifting modules. It is shown that: (1) a module M is ss-lifting iff it is amply ss-supplemented and its ss-supplement submodules are direct summand; (2) for a ring R, RR is ss-lifting if and only if it is ss-supplemented iff it is semiperfect and its radical is semisimple; (3) a ring R is a left and right artinian serial ring and Rad (R) ⊆ Soc(RR) iff every left R-module is ss-lifting. We also study on factor modules of ss-lifting modules. [ABSTRACT FROM AUTHOR]

In this paper we study modules with the WFI+-extending property. We prove that if M satisfies the WFI+-extending, pseudo duo properties and M=(SocM) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the WFI+- extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M = M1 M2 for some semisimple submodule M1 and Noetherian (respectively, Artinian) submodule M2. Moreover, we show that if M is a WFI-extending module with pseudo duo, C2 and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results. [ABSTRACT FROM AUTHOR]

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H -vector (1 , 13 , 12 , 13 , 1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r = 4 , Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H -vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H -vector fails to have WLP. In codimension c = 3 it is conjectured that all AG rings have WLP. For c = 4 , Gondim shows in (Gondim, 2017) that WLP always holds for r ≤ 4 and gives a family where WLP fails for any r ≥ 7 , building on Ikeda's example (Ikeda, 1996) of failure for r = 5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c = 4 and r ≤ 6. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study some special elements (zero divisors and units) in ternary rings. Then, the main properties and concepts of noetherian and artinian ternary rings have been studied. In addition, new results on noetherian and artinian ternary rings have been investigated. [ABSTRACT FROM AUTHOR]

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$. A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized. [ABSTRACT FROM AUTHOR]

Let (C , E , s) be an Ext-finite, Krull-Schmidt and k -linear extriangulated category with k a commutative artinian ring. We define an additive subcategory C r (respectively, C l) of C in terms of the representable functors from the stable category of C modulo s -injectives (respectively, s -projectives) to k -modules, which consists of all s -projective (respectively, s -injective) objects and objects isomorphic to direct summands of finite direct sums of all third (respectively, first) terms of almost split s -triangles. We investigate the subcategories C r and C l in terms of morphisms determined by objects, and then give equivalent characterizations on the existence of almost split s -triangles. [ABSTRACT FROM AUTHOR]

We call a ring R generalized right π-Baer, if for any projection invariant left ideal Y of R, the right annihilator of Yn is generated, as a right ideal, by an idempotent, for some positive integer n, depending on Y. In this paper, we investigate connections between the generalized π-Baer rings and related classes of rings (e.g., π-Baer, generalized Baer, generalized quasi-Baer, etc.) In fact, generalized right π-Baer rings are special cases of generalized right quasi-Baer rings and also are a generalization of π-Baer and generalized right Baer rings. The behavior of the generalized right π-Baer condition is investigated with respect to various constructions and extensions. For example, the trivial extension of a generalized right π-Baer ring and the full matrix ring over a generalized right π-Baer ring are characterized. Also, we show that this notion is well-behaved with respect to certain triangular matrix extensions. In contrast to generalized right Baer rings, it is shown that the generalized right π-Baer condition is preserved by various polynomial extensions without any additional requirements. Examples are provided to illustrate and delimit our results. [ABSTRACT FROM AUTHOR]

Let I and J be ideals of a Noetherian local ring (R,m) and let M be a nonzero finitely generated R-module. We study the relation between the vanishing of HI, Jdim M (M) and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian R-module M/JM is equal to its integral closure relative to the Artinian R-module HI, Jdim M M. [ABSTRACT FROM AUTHOR]

In this paper, we derive some new results on Noetherian and Boolean Artinian BL-algebras. We further obtain some relations between local and semilocal £¡L-algebras and Boolean Artinian BLalgebras. [ABSTRACT FROM AUTHOR]

Given an additive category C and an integer n ⩾ 2. We form a new additive category C ϵ n consisting of objects X in C equipped with an endomorphism ϵ X satisfying ϵ X n = 0. First, using the descriptions of projective and injective objects in C ϵ n , we not only establish a connection between Gorenstein flat modules over a ring R and R t / (t n) , but also prove that an Artinian algebra R satisfies some homological conjectures if and only if so does R t / (t n). Then we show that the corresponding homotopy category K (C ϵ n) is a triangulated category when C is an idempotent complete exact category. Moreover, under some conditions for an abelian category A , the natural quotient functor Q from K (A ϵ n) to the derived category D (A ϵ n) produces a recollement of triangulated categories. Finally, we prove that if A is an Ab4-category with a compact projective generator, then D (A ϵ n) is a compactly generated triangulated category. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and 𝒰R be the set of all nonzero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝒰R and two vertices x and y are adjacent whenever ann(x) ∩ ann(y) = (0). In this paper, we characterize all commutative Artinian non-local rings R for which the 𝒞𝒜R has genus one and two. Also we characterize all commutative Artinian non-local rings R for which 𝒞𝒜R has crosscap one. Finally, we characterize all finite commutative non-local rings for which g(Г2(R)) = g(𝒞𝒜R) = 0 or 1. [ABSTRACT FROM AUTHOR]

Let Q be a (not necessarily unital) simple ring or algebra. A nonempty subset S of Q is said to have zero product if S 2 = 0. We classify all maximal zero product subsets of Q by proving that the map R ↦ R ∩ LeftAnn (R) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Education & Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We use augmented commutative differential graded algebra (Image 1) models to study G -representation varieties of fundamental groups π = π 1 (M) and their embedded cohomology jump loci, around the trivial representation 1. When the space M admits a finite family of maps, uniformly modeled by Image 1 morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of Hom (π , G) and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when M is either a compact Kähler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group G is a non-abelian subalgebra of sl 2 (C). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of c-co-epi-retractable modules. An R-module M is called c-co-epi-retractable if it contains a copy of its factor module by a complement submodule. The ring R is called c-co-pri if RR is c-co-epi-retractable. Conditions are found under which, a c-coepi-retractable module is extending, retractable, semi-simple, quasi-injective, injective and simple. Also, we investigate when c-co-epi-retractable modules have finite uniform dimension. Finally, right SI-rings, semi-simple artinian rings and quasi-Frobenius rings are characterized in termes of c-co-epi-retractable modules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. Let EG(R) be a simple undirected graph associated with R whose vertex set is the set of all nonzero zero-divisors of R and and two distinct vertices x;y in this graph are joined by an edge if and only if AnnR(xy) is an essential ideal. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all Artinian rings whose EG(R) is perfect. [ABSTRACT FROM AUTHOR]

In this article we introduce and study the concepts of α-almost quasi Artinian and α-quasi Krull modules. Using these concepts we extend some of the basic results of α-almost Artinian and α-Krull modules to α-almost quasi Artinian and α-quasi Krull modules. We observe that if M is an α-quasi Krull module then the quasi Krull dimension of M is either α or α + 1. [ABSTRACT FROM AUTHOR]

Abstract We show that on integral normal separated schemes whose function field is separably closed, for each pair of points the intersection of the resulting local schemes is local. This extends a result of Artin from rings to schemes. The argument relies on the existence of certain modifications in inverse limits. As an application, we show that Čech cohomology coincides with sheaf cohomology for the Nisnevich topology. Along the way, we generalize the characterization of contractible curves on surfaces by negative-definiteness of the intersection matrix to higher dimensions, using bigness of invertible sheaves on non-reduced schemes. [ABSTRACT FROM AUTHOR]

Abstract We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1 , d). We consider the algebra associated to polynomials of the same type of bidegree (d 1 , d 2). We prove that the geometry of the Nagata hypersurface of order e is very similar to the geometry of the original hypersurface. We study the Lefschetz properties for Nagata idealizations of order d 1 , proving that WLP holds if d 1 ≥ d 2. We give a complete description of the associated algebra in the monomial square free case. [ABSTRACT FROM AUTHOR]

Abstract Asymptotic properties of high powers of an ideal related to a coherent functor F are investigated. It is shown that when N is an artinian module the sets of attached prime ideals Att A F (0 : N a n) are the same for n large enough. Also it is shown that for an artinian module N if the modules F (0 : N a n) have finite length and for a finitely generated module M if the modules F (M / a n M) have finite length, their lengths are given by polynomials in n , for large n. When A is local it is shown that, the Betti numbers β i (F (M / a n M)) and the Bass numbers μ i (F (M / a n M)) are given by polynomials in n for large n. [ABSTRACT FROM AUTHOR]

In this paper we study rings with ascending (descending) chain conditions on their fuzzy substructures and various results are established. Also we prove some characterizations of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. [ABSTRACT FROM AUTHOR]

Abstract We describe the structure and homological properties of arbitrary generalized standard Auslander–Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic is defined and nonnegative. Further, we provide a handy criterion for an infinite Auslander–Reiten component of an artin algebra to be generalized standard. We solve also the long standing open problem concerning the structure of artin algebras admitting a separating family of Auslander–Reiten components. [ABSTRACT FROM AUTHOR]

Abstract We show that, in the context of Reverse Mathematics, WK L 0 (Weak König's Lemma) implies the statement AR T 0 which says that every Artinian ring is Noetherian, over RC A 0 (Recursive Comprehension Axiom). To achieve this goal, we prove a general Computable Full Structure Theorem for computable Artinian rings similar to the classical version found in most Algebra texts. [ABSTRACT FROM AUTHOR]

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R -modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences. [ABSTRACT FROM AUTHOR]

Artinian quotients R of the local ring Q = k [ [ x , y , z ] ] are classified by multiplicative structures on A = Tor ⁎ Q (R , k) ; in particular, R is Gorenstein if and only if A is a Poincaré duality algebra while R is Golod if and only if all products in A ⩾ 1 are trivial. There is empirical evidence that generic quotient rings with small socle ranks fall on a spectrum between Golod and Gorenstein in a very precise sense: The algebra A breaks up as a direct sum of a Poincaré duality algebra P and a graded vector space V , on which P ⩾ 1 acts trivially. That is, A is a trivial extension, A = P ⋉ V , and the extremes A = (k ⊕ Σ k) ⋉ V and A = P correspond to R being Golod and Gorenstein, respectively. We prove that this observed behavior is, indeed, the generic behavior for graded quotients R of socle rank 2, and we show that the rank of P is controlled by the difference between the order and the degree of the socle polynomial of R. [ABSTRACT FROM AUTHOR]

Abstract A ring R is called left k -cyclic if every left R -module is a direct sum of indecomposable modules which are homomorphic image of R k R . In this paper, we give a characterization of left k -cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe–Cohen–Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k -cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Köthe ring if and only if R is an artinian left multiplicity-free top ring. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to extend the characterization of Teter rings of [9] to Artin rings of Gorenstein colength two and to improve and complete the result [1, Theorem 5.5] on such rings by using Macaulay's inverse system. We provide several explicit examples and families of Gorenstein colength two rings taking into account their analytic type. [ABSTRACT FROM AUTHOR]

We provide a method for constructing central idempotents in the Brauer algebra (using the splitting of short exact sequences of bimodules). From this we determine certain primitive central idempotents. By working over a suitable integral ring we hence demonstrate an efficient method of constructing pieces of the representation theory of the Brauer algebra over Artinian rings from the integral case. [ABSTRACT FROM AUTHOR]

We study the homogeneous artinian ideals of the polynomial ring K [ x , y , z ] generated by the homogeneous polynomials of degree d which are invariant under an action of the cyclic group Z / d Z , for any d ≥ 3 . We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal ( 1 , e , e a ) , where e is a primitive d -th root of the unity. We get a complete description when d is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations. [ABSTRACT FROM AUTHOR]

In this note we extend the main result in [6] on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimal generation of ideals generated by powers of linear forms by the configuration of their dual points in the projective plane and we use this result to improve some propositions on line arrangements and Strong Lefschetz Property at range 2 in [6] . The starting point was an example in [3] . Finally we show the equivalence among failing SLP, Laplace equations and some unexpected curves introduced in [3] . [ABSTRACT FROM AUTHOR]