The Structure of Modules over Hereditary Rings

Let A be a bounded hereditary Noetherian prime ring. For an A-module M _A , we prove that M is a finitely generated projective ${A \mathord{\left/ {\vphantom {A {r\left( M \right)}}} \right. \kern-0em} {r\left( M \right)}}$ -module if and only if M is a ${\pi }$ -projective finite-dimensional module, and either M is a reduced module or A is a simple Artinian ring. The structure of torsion or mixed ${\pi }$ -projective A-modules is completely described.     

Artinian cofinite modules over complete Noetherian local rings

Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0:M p) is not finitely generated and for each i ? 2 the R-module Ext _R ⁱ (M,R/p) is of finite length, then the R-module Ext _R ¹ (M, R/p) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (N,M) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (M,N) are of finite length.     

Zaks’ Lemma for Coherent Rings

Let A be a left and right coherent ring and C _A (resp., $C_{A^{\mathrm{op}}}$ ) a minimal cogenerator for right (resp., left) A-modules. We show that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}}$ whenever flat dim C _A ?<?∞ and $\mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ , and that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ if and only if the finitely presented right A-modules have bounded Gorenstein dimension.     

Free resolutions over short Gorenstein local rings

Let R be a local ring with maximal ideal ${\mathfrak{m}}$ admitting a non-zero element ${a\in\mathfrak{m}}$ for which the ideal (0 : a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when ${\mathfrak{m}^4=0}$ . Let e denote the minimal number of generators of ${\mathfrak{m}}$ . If R is Gorenstein with ${\mathfrak{m}^4=0}$ and e ?? 3, we show that ${{\rm P}_{M}^{R}(t)}$ is rational with denominator H _R (?t) =?1 ? et?+?et ² ? t ³, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.     

Finiteness of graded generalized local cohomology modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring $R = \oplus _{n \in \mathbb{N}_0 } R_n$ with a local base ring (R ₀, m₀) and irrelevant ideal R ₊ of R. We study the generalized local cohomology modules H _b ⁱ (M,N) with respect to the ideal b = b₀ + R ₊, where b₀ is an ideal of R ₀. We prove that if dimR ₀/b₀ ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H _b ⁱ (M,N)/a₀ H _b ⁱ (M,N) is Artinian, where $\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$ ; for all i ≥ 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is asymptotically stable as n→?∞. Moreover, if H _b ⁱ (M,N) _n is a finitely generated R ₀-module for all n ≤ n ₀ and all j < i, where n ₀ ∈ ? and i ∈ ?₀, then for all n ≤ n ₀, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is finite.     

Certain additive maps on m-power closed Lie ideals

Let R be a prime ring with extended centroid C and m a fixed positive integer >?1. A Lie ideal L of R is called m-power closed if ${u^m \in L}$ for all ${u \in L}$ . We prove that if char R = 0 or a prime p?>?m, then every non-central, m-power closed Lie ideal L of R contains a nonzero ideal of R except when dim _C RC?=?4, m is odd, and ${u^{m-1} \in C}$ for all ${u \in L}$ . Moreover, the additive maps d : L ?? R satisfying d(u ^m )?=?mu ^m-1 d(u) (resp. d(u ^m )?=?u ^m-1 d(u)) for all ${u \in L}$ are completely characterized if char R = 0 or a prime p?>?2(m ? 1).     

Homological Behavior of Auslander’s k-Gorenstein Rings

In this paper we mainly study the homological properties of dual modules over k-Gorenstein rings. For a right quasi k-Gorenstein ring Λ, we show that the right self-injective dimension of Λ is at most k if and only if each M?∈ mod Λ satisfying the condition that Ext $_{\Lambda}^i(M, \Lambda)=0$ for any 1?≤?i?≤?k is reflexive. For an ∞-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of Λ are identical. In addition, we show that if Λ is a left quasi ∞-Gorenstein ring and M?∈ mod Λ with gradeM finite, then Ext $_{\Lambda}^i($ Ext $_{\Lambda ^{op}}^i($ Ext $_{\Lambda}^{{\rm grade}M}(M, \Lambda), \Lambda), \Lambda)=0$ if and only if i?≠gradeM. For a 2-Gorenstein ring Λ, we show that a non-zero proper left ideal I of Λ is reflexive if and only if Λ/I has no non-zero pseudo-null submodule.     

A generalization of the finiteness problem of the local cohomology modules

Let R be a commutative Noetherian ring and \(\mathfrak{a}\) an ideal of R. We introduce the concept of \(\mathfrak{a}\) -weakly Laskerian R-modules, and we show that if M is an \(\mathfrak{a}\) -weakly Laskerian R-module and s is a non-negative integer such that Ext _R ^j \((R/\mathfrak{a},H_\mathfrak{a}^i (M))\) is \(\mathfrak{a}\) -weakly Laskerian for all i < s and all j, then for any \(\mathfrak{a}\) -weakly Laskerian submodule X of \(H_\mathfrak{a}^s (M)\) , the R-module \(Hom_R (R/\mathfrak{a},H_\mathfrak{a}^s (M)/X)\) is \(\mathfrak{a}\) -weakly Laskerian. In particular, the set of associated primes of \(H_\mathfrak{a}^s (M)/X\) is finite. As a consequence, it follows that if M is a finitely generated R-module and N is an \(\mathfrak{a}\) -weakly Laskerian R-module such that \(H_\mathfrak{a}^i (N)\) (N) is \(\mathfrak{a}\) -weakly Laskerian for all i < s, then the set of associated primes of \(H_\mathfrak{a}^s (M,N)\) (M,N) is finite. This generalizes the main result of S. Sohrabi Laleh, M.Y. Sadeghi, and M.Hanifi Mostaghim (2012).     

Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$ , W(t, q)?=?a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$ . The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M?>?0 such that (L(t)q, q)?≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$ , we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.     

A criterion for I-adic completeness

Let I denote an ideal in a commutative Noetherian ring R. Let M be an R-module. The I-adic completion is defined by ${\hat{M}^I = \varprojlim{}_{\alpha} M/I^{\alpha}M}$ . Then M is called I-adic complete whenever the natural homomorphism ${M \to \hat{M}^I}$ is an isomorphism. Let M be I-separated, i.e. ${\cap_{\alpha} I^{\alpha}M = 0}$ . In the main result of the paper, it is shown that M is I-adic complete if and only if ${{\rm Ext}_R^1(F,M) = 0}$ for the flat test module ${F = \oplus_{i = 1}^r R_{x_i}}$ , where ${\{x_1,\ldots,x_r\}}$ is a system of elements such that ${{\rm Rad} I = {\rm Rad}\, \underline{{\it x}} R}$ . This result extends several known statements starting with Jensen’s result [9, Proposition 3] that a finitely generated R-module M over a local ring R is complete if and only if ${{\rm Ext}^1_R(F,M) = 0}$ for any flat R-module F.     

Cofiniteness with respect to a Serre subcategory

Let Φ be a system of ideals in a commutative Noetherian ring R, and let be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ . It is shown that if the functor $ D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) $ is exact, then, for any $ \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) $ ∈ for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that $ H_\mathfrak{a}^i (M,N) $ ∈ for all i < t, then $ Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) $ ∈ , provided that is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that $ H_\mathfrak{a}^i (L) $ ∈ , then $ Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) $ ∈ if and only if $ Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) $ ∈ for all j ≥ 0.     

Nilpotent and invertible values in semiprime rings with generalized derivations

Let R be a semiprime ring and F be a generalized derivation of R and n??? 1 a fixed integer. In this paper we prove the following: (1) If (F(xy) ? yx) ⁿ is either zero or invertible for all ${x,y\in R}$ , then there exists a division ring D such that either R?=?D or R?=?M ₂(D), the 2?× 2 matrix ring. (2) If R is a prime ring and I is a nonzero right ideal of R such that (F(xy) ? yx) ⁿ ?=?0 for all ${x,y \in I}$ , then [I, I]I?=?0, F(x)?=?ax?+?xb for ${a,b\in R}$ and there exist ${\alpha, \beta \in C}$ , the extended centroid of R, such that (a ? ??)I?=?0 and (b ? ??)I?=?0, moreover ((a?+?b)x ? x)I?=?0 for all ${x\in I}$ .     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Ascent properties for pairs of modules

Given a flat local ring homomorphism \({R \rightarrow S}\) and two finitely generated R-modules M and N, we describe conditions under which the modules \({{\rm Tor}^{R}_{i}(M,N)}\) and \({{\rm Ext}^{i}_{R}(M,N)}\) have S-module structures that are compatible with their R-module structures.     

Noetherian Rings whose Modules are Prime Serial

A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (?-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\), and we say that M is prime serial (?-serial, for short) if it is a direct sum of ?-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ?-serial?” and “Which rings have the property that every finitely generated module is ?-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring.     

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

Let \(\Lambda = \left( {\begin{array}{*{20}{c}} A&M \\ 0&B \end{array}} \right)\) be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective Λ-modules under the condition that M is a cocompatible (A,B)-bimodule, we establish a recollement of the stable category \(\overline {Ginj\left( \Lambda \right)} \). We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over Λ.     

On the generic kernel filtration for modules of constant Jordan type

Let ${E \cong (\mathbb{Z}/p)^2}$ be an elementary abelian p-group of rank two k an algebraically closed field of characteristic p, and let J =?J(kE). We investigate finitely generated kE-modules M of constant Jordan type and their generic kernels ${\mathfrak{K}(M)}$ . In particular, we answer a question posed by Carlson, Friedlander, and Suslin regarding whether or not the submodules ${J^{-i} \mathfrak{K}(M)}$ have constant Jordan type for all i ≥ 0. We show that this question has an affirmative answer whenever p = 3 or ${J^2 \mathfrak{K}(M) = 0}$ . We also show that this question has a negative answer in general by constructing a kE-module M of constant Jordan type for p ≥ 5 such that ${J^{-1} \mathfrak{K}(M)}$ does not have constant Jordan type.     

REPRESENTATIONS OF AFFINE SUPERALGEBRAS AND MOCK THETA FUNCTIONS

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $ {{\widehat{{s\ell}}}_{2|1 }} $ (resp. $ {{\widehat{{ps\ell}}}_{2|2 }} $ ) can be modified, using Zwegers’ real analytic corrections, to form a modular (resp. S-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N?=?2 (resp.N?=?4) superconformal algebras with central charge 3(1???(2?m?+?2)/M), where m ∈ ?_≥0, M ∈ ?_≥2, gcd(2?m?+?2, M)?=?1 if m?>?0 (resp. 6 (m/M???1), where m ∈ ?_≥1, M ∈ ?_≥2, gcd(2?m, M)?=?1 if m?>?1), whose modified characters and supercharacters form a modular invariant family.     

On one sided ideals of a semiprime ring with generalized derivations

Let R be a ring with center Z(R). An additive mapping ${F : R \longrightarrow R}$ is said to be a generalized derivation on R if there exists a derivation ${d : R \longrightarrow R}$ such that F(xy) = F(x)y + xd(y), for all ${x, y \in R}$ (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and ${F(xy) \in Z(R)}$ , for all ${x, y \in U}$ , unless F(U)U = UF(U) = Ud(U) = (0); (2) ${F(xy) \mp yx \in Z(R)}$ , for all ${x,y \in U}$ ; (3) ${F(xy) \mp [x,y] \in Z(R)}$ , for all ${x,y \in U}$ ; (4) F ≠ 0 and F([x,y]) = 0, for all ${x, y \in U}$ , unless Ud(U) = (0); (5) F ≠ 0 and ${F([x, y]) \in Z(R)}$ , for all ${x, y \in U}$ , unless either d(Z(R))U = (0) or Ud(U) = (0)n.