<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and _S V _R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom _R (I _V (R),?) and Hom _R (?,I _V (R)) exact exact complex \( \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots \) of V-injective modules I _i and I ⁱ , i ∈ N₀, such that N ? Im(I ₀ → I ⁰). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A _V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I _V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if \(Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0\) for all modules I with finite I _V (R)-injective dimension.     

Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Every cluster-tilted algebra B is the relation extension \(C\ltimes \textup {Ext}^{2}_{C}(DC,C)\) of a tilted algebra C. A B-module is called induced if it is of the form M? _C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.     

On finitely generated modules over quasi-Euclidean rings

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let \(\text {E}_n(R)\) be the subgroup of \(\text {GL}_n(R)\) generated by the elementary matrices. In this paper we study the action of \(\text {E}_n(R)\) by matrix multiplication on the set \(\text {Um}_n(M)\) of unimodular rows of M of length \(n \ge k\). Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if \(n > k\). We also prove that \(\text {Um}_k(M) /\text {E}_k(R)\) is equipotent with the unit group of \(R/\mathfrak {a}_1\) where \(\mathfrak {a}_1\) is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.     

Coherence relative to a weak torsion class

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules

$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$

such that F₀,..., F_n?1 are finitely generated free and K_n?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext ⁿ _R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor ^R _n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

     

A natural map in local cohomology

Let R be a Noetherian ring, \(\mathfrak{a}\) an ideal of R, M an R-module and n a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map \(f\colon\operatorname{Ext}^{n}_{R}(R/\mathfrak{a},M)\to \operatorname{Hom}_{R}(R/\mathfrak{a},\mathrm{H}^{n}_{\mathfrak{a}}(M))\), under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.     

Dual automorphism-invariant modules over perfect rings

Under study are the dual automorphism-invariant modules and pseudoprojective modules. Some conditions were found under which the dual automorphism-invariant module over a perfect ring is quasiprojective. We also show that if R is a right perfect ring then a pseudoprojective right R-module M is finitely generated if and only if M is a Hopf module.     

Relative Gorenstein injective covers with respect to a semidualizing module

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every G _C -injective module G, the character module G ⁺ is G _C -flat, then the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is covering.     

Commutative Rings whose Proper Ideals are Serial

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal \({\mathcal {M}}\) such that there exist a uniserial module U and a semisimple module T with \({\mathcal {M}}=U\oplus T\). Moreover, in the latter case, every proper ideal of R is isomorphic to \(U^{\prime }\oplus T^{\prime }\), for some \(U^{\prime }\leq U\) and \(T^{\prime }\leq T\). Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal \({\mathcal {M}}\) containing a set of elements {w ₁,…,w _n } such that \({\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}\) with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results.     

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.     

Bimodule and twisted representation of vertex operator algebras

In this paper, for a vertex operator algebra V with an automorphism g of order T, an admissible V-module M and a fixed nonnegative rational number n ∈1/T Z_+, we construct an A_(g,n)(V)-bimodule Ag,n(M) and study its properties, discuss the connections between bimodule A_(g,n)(M) and intertwining operators. Especially, bimodule A _(g,n)-1T(M) is a natural quotient of A_(g,n)(M) and there is a linear isomorphism between the space IM~k M Mjof intertwining operators and the space of homomorphisms HomA_(g,n)(V)(A_(g,n)(M)  A_(g,n)(V)M~j(s), M~k(t)) for s, t ≤ n, M~j, M~k are g-twisted V modules, if V is g-rational.     

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 lifting modules and strongly dual Rickart modules

The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module has the strongly summand sum property and the generalized Hopfian property, and a ring R is a strongly regular ring if and only if R_R is a strongly dual Rickart module, if and only if aR is a fully invariant direct summand of R_R for every a ∈ R.     

Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, \({\phi\colon R{\to} R}\) is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through \({\phi}\) , then R is regular. This broad generalization of Kunz’s characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If \({\phi}\) is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over \({\phi}\) of any non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.     

On a generalization of semisimple modules

Let R be a ring with identity. A module \(M_R\) is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of \(M_R\) which is a generalization of semisimple and second modules. We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals \(\le \)4 and \(J(R)=0\) is r-semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring.     

An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups

We have defined the weight of the pair (〈S∣R〉,R) for a given presentation 〈S∣R〉 of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of (〈S∣R〉,R) is n, then our algorithm constructs all the n-vertex crystallizations which yield (〈S∣R〉,R). As an application, we have constructed some new crystallizations of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For m≥3 and m≥n≥k≥2, our generalized algorithm gives a \(2(2m+2n+2k-6+{\delta _{n}^{2}} + {\delta _{k}^{2}})\)-vertex crystallization of the closed connected orientable 3-manifold M〈m,n,k〉 having fundamental group \(\langle x_{1},x_{2},x_{3} \mid {x_{1}^{m}}={x_{2}^{n}}={x_{3}^{k}}=x_{1}x_{2}x_{3} \rangle \). These crystallizations are minimal and unique with respect to the given presentations. If ‘ n=2’ or ‘ k≥3 and m≥4’ then our crystallization of M〈m,n,k〉 is vertex-minimal for all the known cases.     

Serre dimension of monoid algebras

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsion-free monoid of rank r and P a finitely generated projective R[M]-module of rank t. Assume M is Φ-simplicial seminormal. If \(M\in \mathcal {C}({\Phi })\), then Serre dim R[M]≤d. If r≤3, then Serre dim R[int(M)]≤d. If \(M\subset \mathbb {Z}_{+}^{2}\) is a normal monoid of rank 2, then Serre dim R[M]≤d. Assume M is c-divisible, d=1 and t≥3. Then P?∧ ^t P⊕R[M] ^t?1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P?∧ ^t P⊕R[M] ^t?1.     

Some results on Artinian cofinite top local cohomology modules

Let \((R,\mathfrak {m})\) be a Noetherian local ring, I be an ideal of R, and M be a finitely generated R-module such that \({\text {H}}_I^t(M)\) is Artinian and I-cofinite, where \(t={\text {cd}}\,(I,M)\). In this paper, we give some equivalent conditions for the property

$$\begin{aligned} {\text {Ann}}\,_R\left( 0:_{{\text {H}}_I^t (M)} \mathfrak {p}\right) =\mathfrak {p}~\text {for all prime ideals }~ \mathfrak {p}\supseteq {\text {Ann}}\,_R{\text {H}}_I^t(M).(*) \end{aligned}$$

Also, we show that if \({\text {H}}_I^t(M)\) satisfies the property \((*)\), then \({\text {H}}_I^t(M)\cong {\text {H}}_{\mathfrak {m}}^t(M/N)\) for some submodule N of M with \({\text {dim}}\,(M/N)=t\).

     

M-$${\mathcal{}}$$-Injective (Flat) and Strongly $${\mathcal{B}}$$B-Injective (Flat) Modules

Let M be a left R-module, \({\mathcal{A}}\)be a family of some submodules of M and \({\mathcal{B}}\)be a family of some left R-modules. In this article, we introduce and characterize \({\mathcal{A}}\)-coherent, \({P\mathcal{A}}\), \({F\mathcal{A}}\), M-\({\mathcal{A}}\)-injective (flat) and strongly \({\mathcal{B}}\)-injective (flat) modules, which are generalizations of coherent, PS, FS, M-injective (flat) and strongly M-injective modules, respectively. We extend some known results to this general structure.     

Non-singular Solutions of Normalized Ricci Flow on Noncompact Manifolds of Finite Volume

The main result of this paper shows that if g(t) is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold M of finite volume, then the Euler characteristic number χ(M)≥0. Moreover, if χ(M)≠0, there exists a sequence of times t _k →∞, a double sequence of points \(\{p_{k,l}\}_{l=1}^{N}\) and domains \(\{U_{k,l}\}_{l=1}^{N}\) with p _k,l∈U _k,l satisfying the following:

1. (i)
   \(\mathrm{dist}_{g(t_{k})}(p_{k,l_{1}},p_{k,l_{2}})\rightarrow\infty\) as k→∞, for any fixed l₁≠l₂;
    
2. (ii)
   for each l, (U _k,l,g(t _k ),p _k,l) converges in the \(C_{\mathrm{loc}}^{\infty}\) sense to a complete negative Einstein manifold (M _∞,l ,g _∞,l ,p _∞,l ) when k→∞;
    
3. (iii)
   \(\operatorname {Vol}_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\rightarrow0\) as k→∞.
    

     

Relative Tor Functors for Level Modules with Respect to a Semidualizing Bimodule

Let R and S be rings and _S C _R a semidualizing bimodule. We investigate the relative Tor functors \(\text {Tor}_{i}^{\mathcal {M}\mathcal {L}_{C}}(-,-)\) defined via C-level resolutions, and these functors are exactly the relative Tor functors \(\text {Tor}_{i}^{\mathcal {M}\mathcal {F}_{C}}(-,-)\) defined by Salimi, Sather-Wagstaff, Tavasoli and Yassemi provided that S = R is a commutative Noetherian ring. Vanishing of these functors characterizes the finiteness of \(\mathcal {L}_{C}(S)\)-projective dimension. Applications go in two directions. The first is to characterize when every S-module has a monic (or epic) C-level precover (or preenvelope). The second is to give some criteria for the isomorphism \(\text {Tor}_{i}^{\mathcal {M}\mathcal {L}_{C}}(-,-)\cong \text {Tor}_{i}^{\mathcal {M}\mathcal {F}_{C}}(-,-)\) between the bifunctors.