Let x¯=x1,…,xk denote an ordered sequence of elements a commutative ring R. M be R-module. We recall the two notions that x¯ is M-proregular given by Greenlees and May Lipman show both are equivalent. As main result we prove cohomological characterization for to in terms Čech cohomology. This implies also M-weakly proregular if it M-proregular. A local-global principle proregularity weakly prov...

In this work, some properties of amply weak essential supplemented modules are investigated. Let M be a projective and weakly module. Then every finitely M-generated module is supplemented. R any ring. if only generated R-module an e-supplement submodule in

We prove that in a ring Rwith an identity there exists an element a ∈ R and a nonzero derivation d ∈ Der R such that ad(a) 6= 0. A ring R is said to be a d-rigid ring for some derivation d ∈ Der R if d(a) = 0 or ad(a) 6= 0 for all a ∈ R. We study rings with rigid derivations and establish that a commutative Artinian ring R either has a non-rigid derivation or R = R1 ⊕ · · · ⊕ Rn is a ring direc...

Fewer forks make faster progress J ust as a large number of forks can inhibit someone unaccustomed to formal dining, an excess of replication forks can slow the rate of DNA synthesis, Zhong et al. reveal. Cdc7 is the catalytic subunit of the Dbf4-dependent kinase, which initiates DNA replication by activating the DNA helicase complex MCM at replication origins. In response to DNA damage, the ch...

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ[M ] such that K ⊕N is weakly injective in σ[M ], for any N ∈ σ[M ]. Similarly, if M is projective and right perfect in σ[M ], then there exists a module K ∈ σ[M ] such that K ⊕ N i...

Let R be a non-domain commutative ring with identity and A(R) be theset of non-zero ideals with non-zero annihilators. We call an ideal I of R, anannihilating-ideal if there exists a non-zero ideal J of R such that IJ = (0).The annihilating-ideal graph of R is defined as the graph AG(R) with the vertexset A(R) and two distinct vertices I and J are adjacent if and only if IJ =(0). In this paper,...

We define two classes of rings calling them weakly clean rings and weakly exchange rings both equipped with the strong property. Although the classes of weakly clean rings and weakly exchange rings are different, their two proper subclasses above do coincide. This extends results due to W. Chen (Commun. Algebra, 2006) and Chin-Qua (Acta Math. Hungar., 2011). We also completely characterize stro...

For a closed symplectic manifold (M,ω), a compatible almost complex structure J , a 1-periodic time dependent symplectic vector field Z and a homotopy class of closed curves γ we define a Floer complex based on 1-periodic trajectories of Z in the homotopy class γ. We show how to associate to the above data an invariant, the symplectic torsion, which is an element in the Whitehead group Wh(Λ0), ...

We show that under minor technical assumptions any weakly nonlocal Hamiltonian structures compatible with a given nondegenerate weakly nonlocal symplectic operator J can be written as the Lie derivatives of J along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamilt...